Bond Risk - Duration and Convexity

8/9/2019 Bond Risk - Duration and Convexity

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1

Bond Risk


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Risks of Fixed Income

SecuritiesOSystematic Risks

O Interest Rate Risk

O Reinvestment RiskO Purchasing Power Risk

O Exchange Rate Risk

OUnsystematic Risks

O Defaut !"redit# RiskO "a Risk

O $i%uidity Risk

&


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'ond Risk ( Rating )genciesO Standard ( Poor*s

O +oody*s

,


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Standard "redit Rating

System

-

Investment Grade:

% .igh /radeAAA - AA Aaa - Aa

% +edium /rade A - BBB A - Baa

Non-Investment Grade:

%

S0ecuative BB - B Ba - B

% Defaut CCC - D Caa - C

Bonds Standard & Poors Moodys

Overall Range AAA - D Aaa - C


5/26

'ond Pricing Formua

!"ou0on onds#

2

1. Semiannual Interest payment = Pmt = 58.75

2. Number of payments to call= n = 5 x 2 = 10

3. Face Value = Future Value = FV = $1000

4. Present Value = Current Price = PV = 1217.50

(must be input as a negative number)

5. Yield to Call = CPT I%= 6.58%


6/26

3

Bond Price and Interest RatesInterest Rate on! Price

".4"# $121%.&'(p &' basis

points) ".*"# $11"'.32 +,$3%.1"-

(p %& basispoints) *.23# $112.2 +,$&&.24-

/o0n &' basis

points) %.*"# $12&.4' +$3".*'-

/o0n %& basispoints) %.%3# $12%.4 +$&".*-


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4

Bond Price and Interest Rates

Price

Interest Rate

Note the asymmetry in the relation.

Bond prices are more sensitie to

drops in interest rates.


8/26

5

Bond A is more interest rate sensitie:better or bot! increases and decreases

in rates. For decreases"increases # in

rates it!ains"%oses# more "%ess#.

Bond A: &i'!er (on)exit*

Bond B: +o,er (on)exit*

Bond B is less interest rate sensitie: or

decreases "increases# in rates it 'ains

"%oses# %ess "more#.

(on)exit*

(on)exit*is a more acc-rate meas-re o interest rate sensiti)it*

t!an -ration. Bond A is more con)ex and se%% at a !i'!er /rice.

ie%d to mat-rit*

Bon

d

Price

ie%d to mat-rit*

Fig. 2


9/26

6

Interest Rate Ris o bonds

Price RiskInerse relation "et#een"ond price and interestrates.

Reinestment Risk$irect relation "et#eencopon renestment incomeand interest rates.

(an be eliminatedb* !o%din' t!ebond to mat-rit*.

$ecreases ininterest rates&aora"le and )ise

)ersa.

(an be eliminatedb* !o%din' eroco-/on bonds.

Increases ininterest rates&aora"le and )ise

)ersa.

3!ese are aected

o//osite%* b*

c!an'es in interest

rates. (an t!e* be

oset4 Yes

3o e%iminate interest rate

ris hold a 'ero copon

"ond to matrity.

3o e%iminate interest rate

ris o&&set price and

reinestment risk.RA3I6


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17

-ration is te point in time in a bon!s life

#here the price risk and reinestment risko&&set each otherso tat te bon! is

immunizedaainst interest rate ris5.

(o) to immni'e a "ond a!ainst interest

rate risk) hold it to its dration.

For a ero co-/on bond d-ration ise-a% to mat-rit*.


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Duration (

Immuni8ationO Duration9 a conce0t deveo0ed y

Fred +acauay in 16,5 that 0rovides

a time:weighted measure of asecurity*s cash ;ows in terms of0ayack

O Immunization9 the conce0t ofminimi8ing the im0act of changes ininterest rates on the vaue ofinvestments

11


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Duration

1&

-rationis a6erae maturity of a coupon bon!.

/uration is a risk*mana!ement strate!yin fi7e!

income in6estment.

Imm-niation strate'* re-ires t!at

ex/ected *ie%d on t!e bond = rea%ied *ie%d


13/26

1,

Rules of Duration)

1. /uration of a 8ero,coupon +no interest payment-

bon! is e9ual to maturity:

2. /uration is ier 0en coupon rate is lo0er.

3. /uration usually rises 0it te bon!s maturity.

4. /uration of a coupon bon! is ier 0en its

;


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1-

. /uration of a portfolio of bon!s is e9ual to te

0eite! a6erae of te !uration of in!i6i!ual

bon!s.

&. coupon bon! can be consi!ere! to be a series

of >ero,coupon bon!s 0ere te maturity of eac>ero is e9ual to te time 0en te cas flo0 occurs:


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12

7. Price c!an'e in a bond :

(!an'e inPrice

is d-ration in *ears* is ie%d to mat-rit*

(!an'e in ie%d tomat-rit* *

PP

=

*"1 9 *#

". The chan!e in price o& a "ond in response to achan!e in interest rates is proportional to itsmodi&ied dration+ lon!er the modi&ied dration)

lar!er the price chan!e.


16/26

13

;odiied -ration"1 9 *#

= .> *ears or >*ear and > mont!s.

At t!is /oint t!e rea%ied *ie%d on t!e

bond ,i%% be e-a% to t!e /romised *ie%d

or sma%% c!an'es in interest rates.

*., "ond is immni'ed a!ainst interest rate

risk i& the holdin! period -inestors decision/

is e0al to its dration.


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Duration

15

?xam/%es:

a. Price c!an'e in t!e 5@*ear bond or a 50 basis

increase in *ie%d to mat-rit* rom .5 to 7.0

* = 0.5 = 0.005

* = 0.5C P = 10D1.17C

= =.1D#"10D1.17#".005#

= @ $21.D "%oss#


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b. Cane in price for a 1' year'ero copon "ondfor a

1'' basis points !ecrease in interest rate or ytm += "#-.

* = 100 basis /oints = 1

= @ 0.01

= =D.2

PP = *D.2#"0.0E2#

= $>D


20/26

&7

10. -ration o a /orto%io o bonds:

/= ,i< i#

,!ere ,i

and i

are t!e /orto%io ,ei'!ts in maret

)a%-e and t!e -ration o t!e it! bond.


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Duration

&1

In)estment trate'*)

When interest ratesare expected to decline, bond

prices increase. !itch to long duration (long

maturit" and lo! coupon rate)bonds. #nd,!hen

interest rates are expected to go up, s!itch to short

duration bonds.

Duration is longer for long maturit"and lo!

coupon bonds, and vice versa.


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Uses for DurationO Providing a measure of a ond*s

voatiity

O Estimating the change in the price ofa bond based on changes in interestrates

O Immuni8ing a ond or ond 0ortfoioagainst interest rate risk

&&


23/26

&,

$ration: s!o,s

c!an'e in bond

/rice a'ainst

interest rate

c!an'es. Note

the linear

relationship.

Cone1ity:Note the

cratre. !o,s t!at

bonds 'ain more in

res/onse to interest

rate decreases and%ose %ess in res/onse

to increases in rates.

$ration s. Cone1ity

Bon

d

Price

ie%d to mat-rit*

Fig. $

Act-a% bond

/rices be!a)et!is ,a*


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&-

(!an'e inPrice

is d-ration in *ears

* is ie%d to mat-rit*

(!an'e in ie%d tomat-rit* *

PP

=

*"1 9 *#

Bond Price c!an'e -sin' -rationG

Bond Price c!an'e -sin' (on)exit*G

PP

=

* 9"1 9 *#

Additiona% term d-e to (on)exit*

12

x Cone1ityx *#2H


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&2

Con6e7ity

12

x Cone1ityx *#2H

*#2is a%,a*s

/ositi)e re'ard%ess o

si'n o

*

* /ositi)eC Bond /rice

dro/s.

(on)exit* term red-ces

/rice dro/ t!ro-'!

*#2and t!e eect is bi''er

or !i'!er (on)exit*.

* ne'ati)eC Bond /rice

increases.

(on)exit* term increases

/rice 'ain t!ro-'!

*#2and t!e eect is bi''er

or !i'!er (on)exit*.

A%,a*s c!oose !i'! (on)exit* Bonds

Cone1ityterm is a%,a*s

/ositi)e and hi!her &or

hi!her Cone1ity


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&3

Cre!it Ris5 Price Ris5 Rein6estment Ris5

Documents

Bond Risk - Duration and Convexity