Lecture (Bonds)

8/9/2019 Lecture (Bonds)

http://slidepdf.com/reader/full/lecture-bonds 1/76

1

An Introduction to Bonds, BondPricing, and Portfolio management

Part: A

Dewan Mostafizur RahmanOctober 2014



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



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



$

• %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



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.



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'



• 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



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

=×+×=

×+×=

+++++++++=



:aluation as the resent :alue of the Bond;s<%

=



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



#$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



&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 .'.=



-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

+

+

≥

−+=+



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.



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



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



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



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



Pricing a Bond :an illustration

The principles of bond pricing can be illustrated using Gilts ()*

Government bonds1'

H&ample 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



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#%$ ++



%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



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



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



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



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



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



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



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



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



E.

2sing a portfolio of 4ero coupon

bonds to replicated a Bond

*aluing a year bond with a coupon rate of 8 per cent.



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.



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



EE

Bootstrapping and )pot (ates

Given

Determine the spot rates for years -# 2# andE'

B t t i d ) t ( t



E$


Given


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



EC


Given


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

=−+−

=−+=

+++==

=

=



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



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'



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'



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-



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



Assessing Bond Price )ensitivit:

.uration and Conve$it



$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



$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



$$

)rice

=ield

)

y

)rice6yield curve

Price -yield relationship



Price sensitivit and bond maturit

)rice

3e$uired =ield y

P

>onger maturity bond

5horter maturity bond



$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



$

)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



$

)rice

=ield

)?

y> y?

)rice6yield curve

Price yield relationship sensitivity

varies with the yield

3elatively low price sensitivity

3elatively high price sensitivity

)>

[]



$=

[].uration: Calculation of the eights applied to the

time of the 8C6



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



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+



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



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

−=

+

−=

+

+

+−=

+

++

+−=∑=



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'



CC

.'-E. .'--E

.'C5

Price

) P#+C t (eiht t =,-




C5


-

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



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



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



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



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



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&.!!



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



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



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$



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+



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



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



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'



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+



Previous #$ample " Add Conve$it

.

onve&ity6 5-55'$-J -E.'26$'-



-

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



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




E


of bonds equals cover duration of liabilit *1+


OIFBIOIT8 F!I/H/



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$# RA#E %ALL$

@




$


of bonds equals cover duration of liabilit *3+


OIFBIOIT8 F!I/H/



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$

@



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



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

=+=

=+=

===+

==

Lecture (Bonds)

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Transcript of Lecture (Bonds)