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Interest Rate Risk II
Chapter 9
Financial Institutions Management, 3/e
By Anthony Saunders
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Price Sensitivity and Maturity
The longer the ter to aturity! the greater
the sensitivity to interest rate changes"
#$aple% Suppose the &ero coupon yield
curve is 'lat at 12(" Bond A pays )1*+2",-
in 'ive years" Bond B pays ),1./"0/ in ten
years! and oth are currently priced at)1..."
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#$aple continued"""
Bond A% P 3 )1... 3 )1*+2",-451"126/
Bond B% P 3 )1... 3 ),1./"0-451"1261.
7o8 suppose the interest rate increases y 1("
Bond A% P 3 )1*+2",-451"1,6/3 )9/+"/,
Bond B% P 3 ),1./"0-451"1,61. 3 )91-"9-
The longer aturity ond has the greater drop inprice"
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Coupon #''ect
Bonds 8ith identical aturities 8ill respond
di''erently to interest rate changes 8hen the
coupons di''er" This is ore readilyunderstood y recogni&ing that coupon
onds consist o' a undle o' &ero:coupon;
onds"
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Price Sensitivity o' +( Coupon Bond
r 8% 6% 4% Range
n
40 )0.2 )1!... )1!2*, )-*1
20 )0+- )1!... )1!1+, )299
10 )919 )1!... )1!.09 )1*.2 )901 )1!... )1!.19 ),*
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Price Sensitivity o' 0( Coupon Bond
r 10% 8% 6% Range
n
40 )020 )1!... )1!2,1 )-.,
20 )0*/ )1!... )1!1-9 )2*-
10 )92, )1!... )1!.0/ )1+22 )901 )1!... )1!.19 ),0
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Rearks on Preceding Slides
The longer aturity onds e$perience
greater price changes in response to any
change in the discount rate" The range o' prices is greater 8hen the
coupon is lo8er"
The +( ond sho8s greater changes in price inresponse to a 2( change than the 0( ond" The
'irst ond is riskier"
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>uration
>uration
Coines the e''ects o' di''erences in coupon
rates and di''erences in aturity" Based on elasticity o' ond price 8ith respect
to interest rate"
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>uration
>uration
> 3 nt31?Ct t451@r6t4 nt31?Ct451@r6t
3 duration
t 3 nuer o' periods in the 'uture
Ct3 cash 'lo8 to e delivered in t periods
n3 ter:to:aturity r 3 yield to aturity"
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>uration
>uration
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Advantages to >uration Measure%
1" Siplicity
2" Can e used to deterine elasticity
et8een price and FTM%
5P4P645r4r6 3 :>?r451@r6
?P451@r6 r 7ote the direct relationship et8een P and :>"
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>uration as Inde$ o' Interest Rate Risk%
The greater the duration! the greater the
price sensitivity and the greater the risk"
Gigher duration indicates that it takes alonger tie to recover the P o' the ond"
This agrees 8ith intuition once 8e reali&e
that H7F a &ero:coupon ond has durationeEual to aturity" A other onds 8ill
have duration #SS than aturity"
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An e$aple%
Consider three loan plans! all o' 8hich have
aturities o' 2 years" The loan aount is
)1!... and the current interest rate is ,("oan J1! is an installent loan 8ith t8o
eEual payents o' )/22"+1" oan J2 is a
discount loan! 8hich has a single payento' )1!.+."9." oan J, is structured as a ,(
annual coupon ond"
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>uration as Inde$ o' Interest Rate Risk
Fieldoan alue 2( ,( P n >
Installent )1.1-"+* )1... )20"90 2 1"-9,
>iscount )1.19"*. )1... ),0"0- 2 2"...Coupon )1.19"-1 )1... ),0"2* 2 1"9*
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iits to >uration Measure
>uration relationship ay not hold i' the
ond has a call or prepayent provision"
Conve$ity"
7egative Conve$ity"
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Special Case and an AdKustent
Maturity o' a consol% M 3 "
>uration o' a consol% >3 1 @ 14R
AdKusting 'or sei:annual payents
dP4P 3 :>?dR451@ 51426R
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Iuni&ing Balance Sheet o' an DI
>uration Lap%
Dro the alance sheet! #3A:" There'ore!
#3A:" In the sae anner used todeterine the change in ond prices! 8e can
'ind the change in value o' eEuity using
duration"
# 3 ?:>AA @ >I R451@R6 or
= [>A: >IkA5R451@R66
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>uration and Iuni&ing
The 'orula sho8s , e''ects%
everage adKusted >:Lap
The si&e o' the DI
The si&e o' the interest rate shock
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An e$aple%
Suppose >A3 / years! >I3 , years and rates
are e$pected to rise 'ro 1.( to 11(" 5Rates
change y 1(6" Also! A 3 1..! 3 9. and # 31." Dind change in #"
= [>A: >IkA?R451@R6
3 :?/ : ,59.41..61..?".141"1 3 : )2".9"
Methods o' iuni&ing alance sheet"
AdKust >A! >Ior k"
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Niitations o' >uration
Hnly 8orks 8ith parallel shi'ts in yield curve"
Iuni&ing the entire alance sheet need not
e costly" >uration can e eployed incoination 8ith hedge positions to iuni&e"
Iuni&ation is a dynaic process since
duration depends on instantaneous R"
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NConve$ity
The duration easure is a linear appro$iation
o' a non:linear 'unction" I' there are large
changes in R! the appro$iation is uch lessaccurate" Recall that duration involves only the
'irst derivative o' the price 'unction"
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NModi'ied duration
P4P 3 :>?R451@R6 @ 51426 CO 5R62or
P4P 3 :M> R @ 51426 CO 5R62
iplies odi'ied duration and CO is aeasure o' the curvature e''ect"
CO 3 Scaling 'actor ?capital loss 'ro 1p rise in
yield @ capital gain 'ro 1p 'all in yield
Coonly used scaling 'actor is 1.0"
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NCalculation o' CO
#$aple% conve$ity o' 0( coupon! 0(
yield! si$:year aturity #uroond priced at
)1!..."CO 3 1.0?P:4P @ P@4P
3 1.0?5999"/,*0/:1!...641!... @
51!..."-+2-,:1!...641!...6
3 20"
