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1
An Introduction to Bonds, BondPricing, and Portfolio management
Part: A
Dewan Mostafizur RahmanOctober 2014
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2
Objectives
• Identify the characteristics of bonds –government and corporate bonds
• Develop a framework for valuing bonds
• Determine the price sensitivity of bonds tointerest rate changes
• Develop the analysis of interest rates and
portfolio management of bonds
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3
Bonds• Bonds are financial instruments that are issued by bodies such as Governments and corporations
to raise funds by promising to pay interest and to repay the amount borrowed at some point in the
future
• The future cash flows promised to the investors in conventional bonds are contractually definedas
– eriodic interest payments
– !epayment of principal
• The value of a bond will be given by the present value of cash flows that can be anticipated bythe holder of the bond
• The rate of return re"uired by investors on a bond# used to calculate the present value of theanticipated net cash flows# will depend on the returns offered by bonds of similar risk and
duration – to understand the benchmark re"uires an analysis of interest rates
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$
• %ace or par value – The amount the issue pays the bondholder at e&piration of the
bond' ()* Government bonds+ ,-..# )/ Government bonds+0-#...'1
• Bond prices are generally "uoted in relation to a par valueof -..
• oupon rate of interest (r c1 – eriodic interest payment on bond (r B1
• 3ero coupon bonds
– These bonds do not make interest payments – The return is in the form of capital appreciation
• 4ther features include risk e&posure# option features (calland convertible provisions1
Bond Characteristics
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5
Valuation of a Bond • The value of a bond will be given by the present value
of cash flows that can be anticipated by the holder ofthe bond
• The rate of return re"uired by investors on a bond willdepend on return offered by bonds of similar risk and
duration'
( ) ( ) ( )
( ) ( ) ( )n
C C C
n
r
B Br
r
Br
r
Br P
r
Principal Payment nterest
r
Payment nterest
r
Payment nterest P
+
+++
+
+
+
=
+
+++
++
+=
1
11
111
20
20
B is the face or par value of a bond
rc is the coupon rate of interest and is contractually defined
r is the prevailing interest rate, the rate that investors can
expect to earn on similar risk investments.
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5
A bond prices as a function of the
required ield Bond price
!e"uired yield
-..
r c6y
.6 f (y1
7hen issued bonds tend to offer a coupon rate ofreturn e"ual to the re"uired rate of return# and
are conse"uently valued at or close to the parvalue'
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• rices and 8ields (re"uired rates of return1 have aninverse relationship
• 7hen yields get very high the value of the bond will bevery low
• 7hen yields approach 9ero# the value of the bondapproaches the sum of the cash flows
Bond Prices and !ields
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8
Valuation of a Bond " An IllustrationA bond, issued 8 years ago with a nominal or face value of £100 promises
to pay a coupon interest payment of 7 per cent on an annual basis and has
12 years to run to maturity. he prevailing interest rate at the time the bond
was issued was 7 per cent, but this has increased to ! per cent today.
he interest yield on comparable ris" bonds being issued today with 12years to run to maturity is ! per cent # this will be the rate of return
re$uired by investors in the bond under consideration.
%etermine its current price.
&78'.8'('''.01001&07.77
)*+100)*A+7)
,0!.01-
1
100,0!.01-
1
7......,0!.01-
1
7,0!.01-
1
7,0!.01-
1
7)
0!.120!.120
1212(20
=×+×=
×+×=
+++++++++=
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:aluation as the resent :alue of the Bond;s<%
=
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10
Price – Interest Rate Relationship
for Bonds
)rice
Interest rate!e"uired rate of return!e"uired yield
) / f -r
0.07 0.0!
100.00
8'.&7
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#$pected Changes in Bond Prices Over
%ime
Time
Price
100.00
85.67
Bond is selling at a discount as the coupon rate is below the prevailing interest rate.
The price of a bond must equal its nominal or face value at the maturity date and its
market value will increase year by year to eliminate the discount by that date.
Maturity date
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&olding'Period (eturn: )ingle Period
>! 6 ? I @ ( - – . 1A / .
where
I 6 r cB6 interest payment
- 6 e&pected price at the end of period one. 6 purchase price
. 6C'5C
- 65'E=5
>! 6 ? I @ ( - – . 1A / .
>! 6 ? @ ( 5'E=5 – C'5C 1A / C'5C 6 .'.=
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-E
#$pected Price Adjustments
t1t
t
t1t
t
c
))where
)
))
)
r 1,-treturnAnnual
≥
−+=+
+
+
Fbove ar
Below ar
apital depreciation
apital appreciation
Time
1tt
t
t1t
t
c
))where
)
))
)
r 1,-treturnAnnual
+
+
≥
−+=+
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14
#$pected annual return on the bond *assuming no change in interest rates over time+
The coupon rate is 7 per cent and the required rate of return
is 9 per cent – therefore the appropriate discount rate.
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Actual changes in bond prices over time
ith unanticipated changes in interest rates+
-C
Bond prices evolving through to maturityBond prices
8ears
onstant re"uired rate of return
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1&
!ield to -aturit
The yield to maturity of a bond is the interestrate that makes the present value of the bond;scash flows e"ual to its price
/olve the bond formula for y (I!!1
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17
!ield to -aturit #$ample: Interest
paid tice a ear
10 yr aturity oupon 3ate / 74
5emi6annual payment / £ (.'0
)ar value / £ 100 urrent mar"et price / £!'
5olve for y / semi6annual rate y / (.8&('4
F bond offers a coupon rate of interest of per cent# with interest being paidon a semiannual basis' If the bond is trading at ,=C'.. and has ten years torun to maturity determine its yield to maturity' ()se H&cel1
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18
!ield -easures
Bond Equivalent Annual Yield
'2 6 E'5 & 2
Effective Annual Yield (Fllows for compounding1
(-'.E512
- 6 'Current Yield
Fnnual Interest J Karket rice
, J , =C 6 'E
alculation of yield
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Pricing a Bond :an illustration
The principles of bond pricing can be illustrated using Gilts ()*
Government bonds1'
H&le taken from -st Karch 2.-2 (at close of trade1
Gilt with a 5 coupon maturing in 2.2'
Luoted price is ,-$2'C
!edemption yield (68TK1 is 2'
oupon interest paid twice a year# th Mune N th Dec
Katures on th December 2.2'
Bond Price Red yield Interest due
Tr 5pc 2.2 -$2'C 2'. Mun J Dec
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Illustration: .etails of a Bond
4n -st Karch the future cash flows from a bond are as follows+
Coupon
,E is due on th Mune 2.-2 (in = days1followed by EE more payments every 5 months until thDecember 2.2
Redemption payment
,-.. on th December 2.2
Oook ahead to ne&t coupon date th Mune 2.-2+
,E @ (,E P F%EE# -'$ 1 @ (-.. P D%EE#-'$ 1
<ow discount back to -st Karch Q -'$
18(.!801.1
01.0((10001.0(((-( "P#$ "P#%$ ++
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%rade price and quoted price
2-
The trade price is ,-$E'=5
>owever# the "uoted price is ,-$2'C'
rices are "uoted after deducting accrued interest# which is theinterest built up since the last payment on Dec 2.--# C days ago'
Fccrued interest 6 -'$.
Luoted price 6 trade price @ accrued interest
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22
)pot (ates and /eros*0+
• F 9ero coupon bond pays no coupon rate ofinterest
• These bonds are issued at a discount to their
face value and the return is obtained in the formof capital appreciation
• These bonds provide the benchmark for theevaluation of cash flows for a particular period –a coupon bond can be viewed as a portfolio of9eros
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2E
/ero Coupon or .iscount Bonds
F 9ero coupon bond# also referred to as a purediscount bond# offers no interest payments# butsimply the payment of the nominal value of thebond at maturity'
Time
. - 2 E $ CRR n
. @Bn
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2$
)pot (ates and /eros*1+
aturity date ime
)rice and nominal value
100
-10
(2.1!7(
<ominal value
F -. year 9ero is issued at a marketdetermined price of ,E2'-=E
Determine the yield
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2C
.eriving the !ield on a /ero "and thereb
the relevant spot rate for that maturit
12.011!7(.(2
100
12.011!7(.(2
100
1/
1010
10
1
10
1
=−==
=−
==
−
=
s
s
n
sn
r y
r y
P
Br y
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25
!ield to maturit and )pot rates
flat. bemustcurveyieldthe
..........
+or ..........
9ne:ttheto periodonefrom
differ mayratesspotthematurity,toyield
they,forvalueoneonlyisthere;hile
(21
(21
sn s s s
sn s s s
r r r r y
r r r r
=====
≠≠≠≠
Katurity
8ieldJ#/pot rates
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2
2se )pot (ates to Value Bonds (-1
%etermine a fair price for a year bond with a coupon rate
of 8 per cent given the following prices and spot rates for
<eros9
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2
2se )pot (ates to Value Bonds (21
%etermine a fair price for a year bond with a coupon rate
of 8 per cent given the following prices and spot rates for
<eros9
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2=
2se )pot (ates to Value
Bonds*3+
*aluing a year bond with a coupon rate of 8 per
cent given spot rates of 10, 11, 11.'0. and 12 per
cent for years one to four.
88.1727
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E.
2sing a portfolio of 4ero coupon
bonds to replicated a Bond
*aluing a year bond with a coupon rate of 8 per cent.
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E-
)pot (ates and !ields
he year bond with a coupon rate of 8 per cent trading
at a price of £88.172! yields 11.88 per cent -a weighted
average of the spot rates.
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E2
2sing )pot (ates to Value Bonds 5ithout
/eros
>ow should bonds be valued in the absence of9eros S
>ow can spot rates be determined using coupon
bondsS
B t t i d ) t ( t
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EE
Bootstrapping and )pot (ates
Given
Determine the spot rates for years -# 2# andE'
B t t i d ) t ( t
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E$
Bootstrapping and )pot (ates
Given
Determine the spot rates for years -# 2# andE'
0!'0.011!!0.11r 1-r
1!!0(.12(!2.8!107r 1-
r 1-
107
1000.1
7&028.!')
10.01100.11(&(&.!7
10&r
2
2s2s
2
2s
2
2s
1s
=−=−+=
==+
++==
=−=−=
B t t i d ) t ( t
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EC
Bootstrapping and )pot (ates
Given
Determine the spot rates for years -# 2# andE'
0!2.010!'.1-'1.1'('01.8!
10'1r 1-r
r 1-
10'0!'.1'1.1'('01.8!)
0!'0.0r
1000.0r
(2
( 2
2s(s
(
(s
2
2s
1s
=−+−
=−+=
+++==
=
=
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E5
6orard rates
The forward rate of interest is the short term (one
period) rate of interest for year n that will make aninvestor indifferent between
investing in a 9ero for n- years and reinvesting theproceeds at maturity in a one year investment and
investing in a n year 9ero
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E
6orard and #$pected 6uture (ates
• The forward rate for some future time period is notnecessarily the short term rate e&pected for this period
• The e&pected rate is often referred to as the e&pectedfuture rate rather than the forward rate
• The relationship between the e&pected future andforward rates depends on the interpretation of the termstructure of interest rates – the yield curve
• If the yield on a two year bond e&ceeds the yield on a
one year bond as a result of a risk premium there is nogood reason to e&pect ne&t year;s one year yield wille&ceed the rate for the coming year'
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E
n−Forward Rates from Observed Spot Rates
Fn investor with a investment hori9on of n years could invest in an year 9ero# or alternatively invest in a n- 9ero and reinvest in a
one year 9ero after n- years' The rate for the nth year that willleave the investor e"ually well off is referred to as the forwardrate'
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E=
Calculating 6orard (ates
$ year spot rate 6 -2 E year spot rate 6 --'C
Determine the forward rate for year $
(-'-21$ 6 (-'--C1E (-@ r f$1
(-'CEC1 J (-'E521 6 (-@r f$1
r f$ 6 '-EC- or -E'C-
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Alternative to ear investment strategies
• F risk neutral investor will be indifferent between the two investmentstrategies
• F risk averse investor with a two year investment hori9on will preferthe two year investment
$.
. - 2
. - 2
Invest in a one year instrument and reinvest the
proceeds at the end of the year for a further year
Invest in a two year 9ero
,-(-@r s-1 -(-@r s-1(-@r 2#s-1
,-(-@r s212
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Assessing Bond Price )ensitivit:
.uration and Conve$it
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$2
• Inverse relationship between price and yield
• Fn increase in a bond;s yield to maturity results
in a smaller price decline than the gainassociated with a decrease in yield
• Oongterm bonds tend to be more price sensitivethan shortterm bonds to interest rate changes
Bond Pricing (elationships
B d P i i ( l ti hi
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$E
• Fs maturity increases# price sensitivity increasesat a decreasing rate
• rice sensitivity is inversely related to a bond;scoupon rate
• rice sensitivity is inversely related to the yieldto maturity at which the bond is selling
Bond Pricing (elationships
*cont7d+
Price yield relationship
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$$
)rice
=ield
)
y
)rice6yield curve
Price -yield relationship
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Price sensitivit and bond maturit
)rice
3e$uired =ield y
P
>onger maturity bond
5horter maturity bond
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$5
• F measure of the effective maturity of a bond – takes into account thetiming of all the cash flows promised by a bond
• It is defined as the weighted average of the time until each paymentis received# with the weights being given by the present value of thecash flow of each period in relation to the price of the bond
• Duration is shorter than maturity for all bonds e&cept 9ero coupon
bonds# where the duration is e"ual to maturity of the bond'
.uration
Price -yield relationship
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$
)rice
=ield
)
y
)rice6yield curve
Price yield relationship
5lope provides the basis ofthe duration measure
5lope measures price sensitivity to yieldchanges
Price -yield relationship – sensitivity
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$
)rice
=ield
)?
y> y?
)rice6yield curve
Price yield relationship sensitivity
varies with the yield
3elatively low price sensitivity
3elatively high price sensitivity
)>
[]
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$=
[].uration: Calculation of the eights applied to the
time of the 8C6
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C.
Propositions relating to duration
• The duration of a 9erocoupon bond e"uals its time tomaturity
• >olding maturity constant# a bond;s duration is higherwhen the coupon rate is lower
• >olding the coupon rate constant# a bond;s durationgenerally increases with its time to maturity
• >olding other factors constant# the duration of a couponbond is higher when the bond;s yield to maturity is lower
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C-
.uration and the Price )ensitivit of a Bond to
Changes in the (equired !ield
( ) ) y*+1
+n)B *
) y*+1
Br +n) *&&&&*
) y*+1
Br +2) *
) y*+1
Br +1)
y ,
y'
'P
) y*+1
+-n)B *
) y*+1
Br +-n) *&&&&*
) y*+1
Br +-2) *
) y*+1
Br +-1) ,
y'
'P
)n B+1 * y ) B+1 * yr n&&&& ) B+1 * y2 r ) B+1 * y1r y'
'P
) y* B+1* ) y* B+1r *&&&&* ) y* B+1r * ) y* B+1r P
) y*+1
B *
) y*+1
Br *&&&&*
) y*+1
Br *
) y*+1
Br P
nn
ccc
1*n1*n
c
.
c
2
c
1)+n1)+n
c
2
C
2
C
nn
ccc
nn
ccc
2
21
2
,1-
1
,1-
,1-
,1-
+−
+
+
−−−−−=+
=
=
+−+−−−
−−−−
onsider the derivative of the price of a bond in relation to its yield+
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C2
-acaula7s .uration
( )
( )
Duration Macaulay/s P
y) 0C$+1t "
y) y)+1'+1
'PP
P ) y*+1
+n)B *
) y*+1
Br +n) *&&&&*
) y*+1
Br +2) *
) y*+1
Br +1)
, y y'
P 'P
) y*+1+n)B *
) y*+1 Br +n) *&&&&*
) y*+1 Br +2) *
) y*+1 Br +1)
y ,
y' 'P
t
nn
ccc
nn
ccc
=+
∑−=++
−++
+−
+
2
2
,1.-,1-.
,1-1
,1-
The proportionate change in the price in relation to a ver small proportionatechange in the yield factor
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CE
-acaula7s .uration and -odified .uration*3+
Duration Mo'ifie'
y
Duration Macaulay/s
P
"
y'
'P
uration Mo'ifie' Dre' to as y)is refer + factor yiel' theto Duration Macaulay/sof ratio!he
Duration Macaulay/s " P
" y'
'P
by i3enis priceinchane percentaeeappro"imat !he
P
nB Br t
MacD
Duration Macaulay/s
n
t
C
1-
1
1-
1
y-1
11
1-
y-1y-1
-
-t
1t
−=
+
−=
+
+
+−=
+
++
+−=∑=
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Calculating .uration
C$
F three year bond offering a coupon of -C per cent hasa yield of -C per cent and is selling at par'
The bond;s duration is calculated to be 2'525'
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CC
.'-E. .'--E
.'C5
Price
) P#+C t (eiht t =,-
Calculating .uration
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C5
Calculating .uration
-
2
E
.'-E.
.'--E
.'C5
.'-E.
.'225
2'25
D 6 - & (-CJ-'-C1J-..@ 2 & -CJ-'-C21J-.. @ E & --CJ-'-CE1J-..6 - & .'-E.$ @ 2 & .'--E$ @ E & .'C5- 6 2'525
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C
-odified .uration
y " Duration Mo'ifie' 4PP
y) +1 Duration Macaulay/s Duration Mo'ifie'
P
y) 0C$+1t "
y) y)+1'+1
'PP Duration Macaulay/s
t
∆−=
+−=
+∑−=
++
−=
The proportionate change in the price in relation to the proportionatechange in the yield factor
2sing .uration
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C .&5 1.0&52 " 0&006 "1&10
7&
y " P " y1
Duration Macaulay/s priceChane in 8stimate'
7&1.0&52
P
y) C$+1t "
y) y)+1'+1
'PP Duration Macaulay/s
t
(222812
281208.821
−=−=
∆+
−=
==
+∑=++
=
Fssume re"uired yield increased from -. to -.'C per cent
Analsing the impact of changes in the ield
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C=
Analsing the impact of changes in the ield
There is a -C per cent coupon bond trading at -E.'2 to provide a
yield of -. per cent' 7hat will be the impact of the re"uired yieldincreasing to -.'C. per
Using Duration
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5.
Using Duration
(22(.71(0.72: 0.00' :
1.10
812&.2
y : ):y1
%urationacaulay@s pricehange instimated
812&.21(0.72
081!.218
)
y B+-1t :
yy-1d-1
d)) %urationacaulay@s
t
−=−=
∆+
−=
=−=
+∑−=++
−=
Fctual change in price+-E.'2-2'. 6 E'5C
Hstimated change in price+E'EDifference6.'.
Determining Duration
Errors in using duration to determine the effects of a
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5-
)rice
=ield
)0/1(0.72
y0
)rice6yield curve
g
yield change
rror in estimating price change usingduration / 0.08
y1
)1/127.07
st.)1/12&.!!
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Assessing .uration
• Kodified duration provides a reasonable basis forestimating the proportionate change in price for smallchanges in the re"uired yield
• 7hen there are more significant changes in the
re"uired yield# modified duration fails to provide anade"uate basis to estimate the price change'
• Duration will overestimate the price change when there"uired yield increases# thereby underestimating the
new price'• 7hen the re"uired yield falls# duration willunderestimate the price change and therebyunderestimate the new price'
52
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Portfolio .uration
• /o far we have looked at the duration of an individual bond'• The duration of a portfolio is simply the weighted average
duration of the bonds in the portfolios'• ortfolio managers look at their interest rate e&posure to a
particular issue in terms of its contribution to portfolio duration'
• This measure is found by multiplying the weight of the issue inthe portfolio by the duration of the individual issue given as+
contribution to portfolio duration 6 eight of securit in portfolio
P duration of securit
5E
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Conve$it and ield changes
• Fs we have seen the first derivative can be represented by atangent line is drawn to the price–yield relationship at yield 9
• The tangent shows the rate of change of price with respect to achange in interest rates at that point (yield level1'
• Because all the duration measures are only appro&imations forsmall changes in yield# they do not capture the effect of thecurvature of the priceyield relationship
• The curvature is referred to as the bond;s conve&ity and when yieldschange by more than a small amount this needs to be taken into
account'• The duration measure can be supplemented with an additionalmeasure to capture the curvature or conve&ity of a bond to improvethe estimated changes in price as the yield changes'
5$
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5C
onve&ity (a1
he first steps in deriving the conve:ity of a bond is todetermine the second derivative of the price of a bonds with
respect to its yield9
) y*+1
B *
) y*+1
Br *&&&&*
) y*+1
Br *
) y*+1
Br , P
nn
c
2
c
1
c
) y*+1
+-n)B *
) y*+1
Br +-n) *&&&&*
) y*+1
Br +-2) *
) y*+1
Br +-1) ,
y'
'P 1*n1*n
c
.
c
2
c
,1- +
C it *b+
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55
Conve$it *b+he second derivative is given by
∑
∑
=+
=
++
++=
++
+=
+
++
+
+++
++
++=
+−+−−−−+
n
1t2tt
y)(1
11)NCFt(t
n
1ttt2
2n2n
c
2
cc
2
2Cn2Cn
c
c
(
c
2
2
y1-
1 B+1t-t
y1-
1
y1-
1n-n
y1-
r 1n-n
y1-
r &
y1-
r 2
y1-
1
yC-1
1n--6n C
yC-1
r 1n--6n C....C
yC-1
r (--62 C
yC-1
r 2-1- /
y1-d
)d
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5
onve&ity (c1
onve:ity is given by 12 the second derivative divided
by the price of a bond
∑
∑
=+
=
++=
++
+=
n
t t t
n
t t t
P y
C$ t t
P y
C$ t t y
12
12
1-
11-21
1-
11-
1-
1
2
1
-easuring -acaula7s .uration
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5
-easuring -acaula s .uration
and Conve$it F -2 per cent coupon bond is trading at --C'=. to yield per cent'Determine its duration and conve&ity'
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5=
Calculating Conve$it
1781.2!708.11'00'.8(7&.!00'.08.111.-
8(7&.!
!708.11'7&'.22812
1
)rice%erivative-2
1
bygivenis
2 −=+−=∆
==
=
" " " P
"
9econ' "
Con3e"ity
/econd Derivative+
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Previous #$ample " Add Conve$it
.
onve&ity6 5-55'$-J -E.'26$'-
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-
Invest to cover future liabilit
. - 2 ERRRRRRRR''n RRRRRRRRR''m
Invest
!eceive interestand reinvest
/ell original bonds
plus bonds bought
using interest income
)<H!TFI< /)K
OIFBIOIT8 F!I/H/
Liaility in year n! onds mature in year m
B4<D/ KFT)!H
Immunisation strateg eighted average duration
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2
Immunisation strateg " eighted average duration
of bonds equals cover duration of liabilit
. - RRR''n RRRRRRRRR''m
Invest
!eceive interestand reinvest' Flso reinvestthe proceeds from the saleof maturing bonds
/ell original bondsplus bonds bought
OIFBIOIT8 F!I/H/
Oong DurationBonds Kature
/hort DurationBonds Kature
Immunisation strateg eighted average duration
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E
Immunisation strateg " eighted average duration
of bonds equals cover duration of liabilit *1+
. - RRR''n RRRRRRRRR''m
OIFBIOIT8 F!I/H/
Oong DurationBonds Kature
/hort DurationBonds Kature
Interest rate falls – additionalbonds bought are more e&pensiveand interest income generated islower but sold at higher price at time n
Interest rate falls – higher pricereceived for long maturity bondsat time n
I"#ERE$# RA#E %ALL$
@
Immunisation strateg eighted average duration
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$
Immunisation strateg " eighted average duration
of bonds equals cover duration of liabilit *3+
. - RRR''n RRRRRRRRR''m
OIFBIOIT8 F!I/H/
Oong DurationBonds Kature
/hort DurationBonds Kature
Interest rate rises – additionalbonds bought are cheaper and interest income generated ishigher but sold at lower price at time n
Interest rate rises – lower pricereceived for long maturity bondsat time n
I"#ERE$# RA#E RI$E$
@
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Immunisation Illustration
• F pension fund has a liability of ,-. millionto meet $ years from now
• The yield curve is flat and the interest rate
is 5 per cent
• There are two 9ero bonds available forinvestment# one matures after two years
and the other after years• onstruct an immunised portfolio
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Immunisation Illustration ' solution
[ ]
[ ] 0''''
1
years.aftermoreworth bewill<erosyear8originalthe
therateinterestlowerawith?owever,<ero.year8in theinvestedyearstwo subse$uentfor thecent per'.'earnonlywill but thissame,theremains
<eroyear2thefrom payoff cent the per'.'tochangesinteresttheDf
1
lyrespectivem£'.281andm£2.& ie
<eroyear2in thethirdstwoand<eroyear8in thethirdoneDnvest
1( w2&w 8ww6-1:2
22
8
22
8
m10 )+1&&07)6&2:1 " +1)+1&0
07)2&74 m +1&#
m10+1&07)&07)6&2:1 " +1+1&07)
07)2&74 m +1&#
(eihts
5&;21 m+1&07)
1
10 "m P# of <10
4
4
4
=+=
=+=
===+
==