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Introduction to the
Measurement of InterestRate Risk
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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.
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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.
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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
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(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.
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(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 $ond isapproimatel! the same -#hether increase ordecrease in !ield for a small chan%e in !ield.
• For a lar%e chan%e in !ield the percenta%e pricechan%e increase in %reater that the percenta%eprice decrease.
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(rice Volatilit! of 5onds #ith :m$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
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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! $e retired at the call price.
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5onds #ith :m$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.
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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
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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.'"+
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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
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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$edded
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Macaula! Duration
It is calculated as)*acaulay %uration # *od. %uration (& + yield)
-here,
# num$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.
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Interpretations of duration
1
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(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.
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(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+
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Conveit! measure
It is used to approimate the chan%es in price thatis not eplained $! duration.
Convexit! *easure # C x +,! - ./ x 011
<here
G!J @ the chan%e in !ield for #hich the percenta%e
price chan%e is sou%ht.
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Modi&ed conveit! and eHective conveit!
*odi2ed convexit!• It is the conveit! measure investors o$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.
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(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.
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