FINM7405 Interest Rate Risk 2011-W1

8/6/2019 FINM7405 Interest Rate Risk 2011-W1

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FINM7405

Kam Fon Chan

Interest Rate Risk Management

University of Queensland Business School


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Lecture outline:

Types of short-term and long-term debt11

This is an example text. Go ahead and replace it33

Theories of interest rates term structure

Fundamentals of bond pricing44

55

66

Bond price volatility

77 Duration, DV01 & hedging


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Summary:

There are various short- and long-term debts inAustralia.

There are 4 main yield curves/term structureo Normal (upward sloping) curveo Flat curveo Inverse (downward sloping) curveo Humped curve

The different yield curve shapes can be explained by 3different theorieso Pure expectation theoryo Liquidity theoryo Market segmentation theory


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Summary:

When we calculate bond price, we obtain clean price.What investors really pay is the dirty price.

Yield to maturity (aka internal rate of return) is a constantrate that makes the resent value of the bond e uals toits market price

maximize/minimize bond price volatility. There are 2duration measures

o acau ay ura ono Modified duration


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Summary:

Duration can be used as an approximation to interestrate sensitivity of a bond (ie how the bond price will reactto a very sma c ange n t e y eo How to approximate bond price change? Compute DV01 aka

PV01o y mere y approx ma on ecause o convex y e ec

If we hold (long position) bond B, we can perfectly hedgeit by taking a short position in bond H.o How? By making the portfolio DV01 = 0


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Short-term and long-term debt in Australia

Long-term (maturity up to 30years)

Short-term (maturity up to12 months)

Bank Accepted Bills (BAB) Treasur notes

Treasury bonds State bonds

Certificates of Deposit (CD) Commercial Papers Interbank deposit

Corporate bonds Eurobonds


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Bank Accepted Bills (BAB)

Similar to a cheque, but with fixed maturity of 90, 120or 180 da s

Involve 3 parties: drawer, discounter & acceptor

Drawer Discounter

Acce tor

SellBABi.e.borrowcash

PromisetopayfacevalueofBAB(e.g.,


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Drawer Discounter

Acce tor

BuyBABi.e.lendcashtodrawer

Cansell(trade)BABtoanotherdiscounter Finalholder discounter receivesfacevalue

atmaturityofbill


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Drawer Discounter

Acce tor

Thefacilitator(i.e.bank) Guarantee(accept)theBABi.e.guaranteethedrawerwillpaythe

facevalueonmaturit b takin thecashfromthedrawerand assit

tothediscounter


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Treasury notes

Issued by the Australia gov. to assist in within-yearfundin needs.

Within-year funding needs arise because timing of gov.revenues does not match expenditure profile.


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Certificates of deposit (CD)

Usually matures between 1 and 3 months Issued b banks to raise funds to finance their lendin

activities Contain credit risk of the issuing bank


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Commercial Papers

Commercial papers (CP) are unsecured short-term debt Usuall matures between 2 and 270 da s

Instead of taking bank loans, companies with high creditrating issue CP to raise funds to finance projects

reflect interest


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Interbank deposit

Most popular is LIBOR (London Interbank Offer Rate)

o LIBOR is the interest rate at which banks offerto lendunsecured funds to another bank in the London interbankmarket

o Usually matures between 1 and 90 dayso Singapore Interbank Offer Rate (SIBOR) is the LIBOR-

equivalent bank offer rate in Singapore interbank marketo ur or s e equ va en an o er ra e n uro n er an

market.o Daily LIBOR rate is determined by the British Bankers

average of the rates supplied by member banks (seehttp://www.bbalibor.com/bbalibor-explained/the-basics)

Yen and Swiss francs.


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Interbank deposit

Most popular is LIBOR (London Interbank Offer Rate)

o Why is LIBOR so popular?

Widely used as a reference rate for interest rate swaps etc

Commonly used as a proxy for risk-free rate in practice (but

seldom in academic)


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Treasury bonds

Issued by Australia gov. T icall matures in 2 10 rs

Has face value and pays coupons, payable every 6months

(but not in practice)


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State bonds

Semi-gov bonds issued by State Treasury (e.g. QTC) tomeet fundin needs of state, local ov and ov

instrumentalities (to build new port etc).


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Corporate bonds

Issued by companies T icall maturit is u to 10 rs

Has face value and pays coupons, usually payableevery 6 months


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Eurobonds

Eurobond is a bond issued in a currency (eg. USD)other than the currenc of the countr e . Australia or

market (eg. Japan) in which it is issued. Eurobonds are classified based on currency in which the. . ,

bonds

Example: A Eurodollar bond issued in Japan by anustra an company s a uro on

Attractive because issuer (eg. Australian company) canchoose the countr (e . Ja an) in which to offer its bond

in its preferred currency (eg. USD)


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Shapes of interest rates term structure

The interest rates term structure/yield curve at time tdefines the relationshi between the level of interest

rates and their time to maturity

o Normal (upward sloping) curveo Flat curve

o Humped curve

The sha e of term structure serves as an indicator of

market expectation towards direction of future interestrates (see Figure 1)


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Normal (upward sloping) curve Yieldfor15yr=3.99%p.a.

Yieldfor1yr=0.49%p.a.

Figure 1: U.S. treasury curve on June 10, 2010. Source: Bloomberg


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Flat curve, which usually leads to .

gure : apanese governmen y e curve on ov. , . ource: oom erg


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Inverted (downward sloping) curveYieldfor3mth=6.11%p.a.

Yield for 10 r = 5.31% .a.

gure : . . reasury curve on ec. , . ource: oom erg


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Humped curve

gure : . . reasury curve on ay , . ource: oom erg


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Two points to notice: Different dates have different shapes of term structure

FlatinAug2000

InvertedinFeb2001

Figure 5: U.S. dollar swaps curve between Aug 29, 2000 and Aug 29, 2001. Source: Bloomberg


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Inverted term structure signals economic recession. Proof:

16

183-month T-bill

10-year bond

8

10

12

14

rcentage

2

4

6P

0

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Apr/01

Apr/04

Apr/07

Figure 6: 3-month U.S. T-bill vs. 10-year U.S. treasury bond. Source: H.15database released by U.S. Federal Reserve.


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Inverted term structure signals economic recession. Proof:

4

5

1

2

ercentage

-2

-1

0

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Apr/56

Apr/59

Apr/62

Apr/65

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Apr/71

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Apr/77

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Apr/83

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Apr/95

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Apr/01

Apr/04

Apr/07

P

-3

Figure 7: U.S. spread (10-year treasury bond minus 3-month T-bill). The shaded areas. . . .

released by U.S. Federal Reserve and NBER.


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Theories of interest rates term structure

The different shapes of interest rates term structure canbe ex lained b 3 different theories:o Pure expectation theoryo Liquidity preference theory

(See attached reading by Fabozzi F., (2007 2nd ed.), Fixed Income Analysis,

John Wiley & Sons, Inc., pp. 79-81)


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Fundamentals of bond pricing

Standard formula to price BAB:

)1(1

+

= tFVPBAB

365

where FV = face value of bill

y = yield

t = days to maturity


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Example:o On 2 Feb 2010, Hi hGear issued a $100,000 90-da

BAB with 9.5% p.a. yield to LowGear. What is the priceof the bill?

000,001$ ,

365

90095.01

+

BAB


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Example (Cont):o 30 days have passed. On 4 Mar. 2010, LowGear sold

the BAB to TopGear at a new yield of 8.5% p.a. What

is the price of the bill?

You try over here!

Time 0 30 days later 90 days later

LowGear lent$97,711 toHighGear

LowGear sold theBAB to TopGearat $98,622

TopGear redeemed theBAB from HighGear i.e.HighGear paid $100k toTo Gear

LowGear gained $911 TopGears interest (received) = $1378


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Standard formula to price a coupon-paying bondassumin discrete com oundin :

11 yn

+

)2(

1m

ym

C

m

y

FVP

nC+

+

=

where FV = face value of the bondm isusually2because:

C = coupon amount (pa)

y = yield (pa)

n = number of periods

coupons

are

paid

semi

annually accordingly,yieldsarecompoundedsemiannually

m = compounding frequency


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Example:o A 5.3% .a. semi-annual cou on Treasur bond

maturing in 2 years is priced at 6% p.a. compoundedsemi-annually. The bond has a face value of $1mil.

.

4

( )990,986$

206.0

2.

2

000,53$

206.01

000,000,1$4

=

++

+=CP


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Example (cont):o A 5.3% .a. semi-annual cou on Treasur bond

maturing in 2 years is priced at 6% p.a. compoundedsemi-annually. The bond has a face value of $1mil.

. tm

m

y

+1Assumecouponsandfacevalueare

stripped into

4

zero

coupon

bonds

with

Time to

maturity (1)

Cash flow (2) Discount factor (3) Present value (4) =

(2) x (3)

0.5 26500 0.9709 25728

1.0 26500 0.9426 24979

1.5 26500 0.9151 24251

2.0 1026500 0.8885 912032

Sum 986990


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Example (Cont):o 6 months have assed i.e. the Treasur bond now has

1.5 years to maturity. The current yield is 5% p.a.compounded semi-annually. Calculate the bond fair

.

You try!


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Example (Cont):o Another 4 months have assed i.e. the Treasur bond

now has 1 year & 2 months to maturity. The currentyield is 4.8% p.a. compounded semi-annually.

.

You try!


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Example (Cont):o Another 4 months have passed i.e. the Treasury bond

now as 1 year & 2 mont s to matur ty. e current

yield is 4.8% p.a. compounded semi-annually.Calculate the bond fair rice.

t=1yr2mthst=8mthst=2mthst=4mths

= = t=


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Example (Cont):o Another 4 months have passed i.e. the Treasury bond

now as 1 year & 2 mont s to matur ty. e current

yield is 4.8% p.a. compounded semi-annually.Calculate the bond fair rice.

t=1yr2mthst=8mthst=2mthst=4mths

= = t=


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Discrete compounding vs continuous compounding

y=4.8%p.a.

compoundedsemi

)3(1ln

+=

my

mr


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Example (again):o Another 4 months have assed i.e. the Treasur bond

now has 1 year & 2 months to maturity. The currentyield is 4.8% p.a. compounded semi-annually.

.tr

edf=

Time to

maturity (1)


(2) x (3)

. . ,

0.666667 26500 0.968873 $ 25,675

1.166667 1026500 0.946165 $ 971,238

Sum $1,023,205


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Summary:o The bond rice is the same re ardless if ou use

discrete compounding (eg 4.8% pa compounded semi-annually) or continuous compounding (eg 4.7433% pa


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Advanced issues in bond pricing

Clean price vs. dirty priceo Clean price (quoted in Bloomberg system etc) is like an

o What investors actually pay is dirty price:

Dirtyprice=Accruedinterest+cleanprice


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Example:o Another 4 months have passed i.e. the Treasury bond now has 1

. . . .compounded semi-annually. Calculate the bond fair price.

t=$26500t=$26500 t=$26500+$1mil

couponlastsincediffdayCinterestAccrued =

now

667,17$6

426500$

==

o Dirty price = $1,023,305 + $17,667=$1,040,872


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Term structure is not flato A coupon-paying bond can be stripped into nzero-coupon bondo Each nth zero-coupon bond has its own discount rate and time to

maturityo Thus, discount each zero-coupon bond using its own discount

ra e me o ma ur y an sum up o ge e c ean on pr ce

o Example: A 5.3% p.a. semi-annual coupon Treasury bond. -

is upward sloping (see next slide). Calculate the bond fair cleanprice.


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tm

m

y

+1

Time to

maturity (1)

Yield (p.a.)

(2)

Cash flow (3) Discount factor

(4)

Present value

(5) = (3) x (4)

0.5 4.6% 26500 0.9775 25,904

1.0 5.6% 26500 0.9463 $ 25,076

1.5 5.8% 26500 0.9178 $ 24,322

2.0 6.0% 1026500 0.8885 $ 912,032

Sum = $987,334


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Yield to maturityo Internal rate of returno A constant rate that makes the resent value of future cash flows

equals to the current market priceo Think of it as another way to re-express the bond price

tm

m

y

+1

Whatisythat

solvesforbond

price=$987,334?

Time to

maturity (1)


(2) x (3)

0.5 26500 ? ?1.0 26500 ? ?

1.5 26500 ? ?

2.0 1026500 ? ?

Sum $987,334


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In summaryo Pricing bond using spot rate and yield to maturity gives the same

result.


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Bond price volatility = percentage change in bond price Some useful relationshi s:

a. Bond prices are inversely related to yields

b. Bond price volatility is positively related to term to maturityc. Bond price volatility increases at a diminishing rate as term to

maturity increasesd. A decrease in yield raises bond prices by more than an

increase in yield of the same amount lowers prices (eg if a 1%ecrease n y e ra ses on pr ce y , en aincrease in yield will lower bond price by only $9)

e. Bond price volatility is inversely related to coupon


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Trading strategieso Assume you were an asset managero You predict a major decline in yields you predict an

increase in bond prices (#a) You want a portfolio of bonds with maximum bond price

vo a y o en oy max mum pr ce c anges cap a ga nsfrom changes in yields

You should buy long-term maturity bonds with low

D i DV h d i


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Duration, DV01 & hedging

Savings and loan debacle in 1980so U.S. savings and loan companies earned a spread between

long-term mortgage rates and short-term deposit rates

o Positive spread (ie profit) if long-term mortgage rates > short-term deposit rates

o u n ear y s

16

18

4

6

8

10

12

14

Percent(%)

0

2

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Jan/82

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Jan/86

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Jan/94

Jan/96

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o Implication: Interest rate risk management is important!

-mon . . - ource: . a a ase re ease y . . e era eserve.

D ti DV01 & h d i


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How to measure and manage interest rate?

Duration

DV01 Futures/forwardValueatRisk(VaR) wap

D ti DV01 & h d i


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Duration:o Bond price volatility is positively related to term to maturity

(#b) but inversely related to coupon (#e)

o Need a composite measure to combine #b and #e tomaximize/minimize bond price volatility

o e compos e measure o on pr ce vo a y s ura on

Duration DV01 & hedging


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Characteristics of duration:o Duration of zero-coupon bond = term to maturityo Duration of coupon bond < term to maturity

o Duration is inversely related to coupon rateo Duration is positively related to term to maturityo ura on s nverse y re a e o y e o ma ur y

Two duration measures:o Macaulay durationo Modified duration



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Macaulay duration (example)o A 5.3% p.a. semi-annual coupon Treasury bond maturing in 2

years is priced at 6% p.a. compounded semi-annually. The

bond has a face value of $1mil. Calculate the Macaulayduration of the bond

Time to

maturity (1)

Cash flow (2) Discount

factor (3)

PV of cash

flow (4)

Weight (5) Time x Weight

(1) x (5)

. . 25,728 0.02607 0.0130

1.0 26500 0.9426 24,979 0.02531 0.0253

1.5 26500 0.9151 24 251 0.02457 0.0369

2.0 1026500 0.8885 912,032 0.92405 1.8481

Sum 986,990 1 1.9233

1.92yrs



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Macaulay duration = 1.92 years

Modified duration (in years): )4(

1 +

=

my

ModD

867.106.0

9233.1=

=ModD

2



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Interpretation:o Not helpful to think of duration in terms of timeo Better interpretation: The bond price is sensitive to rate

changes of a 1.867-year (modified duration) zero-couponbond, or

o e on pr ce w approx ma e y c ange y . or abasis point change in the yield

100bp =1%



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Macaulay & modified durations (you try!)o A 7% p.a. semi-annual coupon bond maturing in 5 years has a

yield to maturity of 8% p.a. compounded semi-annually. The

bond has a face value of $1mil. Calculate its Macaulay andmodified durations



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Why use Macaulay/modified duration?o Provide a price approximation to interest rate sensitivity of the

bond (with no embedded options eg not a convertible bond) ie

how bond price will react to a very small change in yieldo How?

Property#a

)5(yPodDP =

DV01(dollarvalueof1basispoint)orPV01(price

va ueo as spo nt e ow on pr cew c ange

iftheyieldchangesby1bp?



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Examples:o Calculate the DV01:

30.184$0001.09869908673.1valueabsolutein

P=

=

%018673.0

986990

.==

P

o Full valuation: A 5.3% p.a. semi-annual coupon Treasury bond

.

Yield(compounded semi Bondprice Difference

5.99% $987,174 +$184

6.00% $986,990 NA

6.01% $886,806 $184



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Remember:o Modified duration only provides a price approximation to a

very small change in yield (eg 1 bp change in yield)

o Example: Use modified duration to calculate priceapproximation when the yield changes by 50 bp

)(9215$

0050.09869908673.1

valueabsolutein

P

=

=

o Full valuation:

annually)

5.50% $996,261 +$9270

. ,

6.50% $977,830 $9160



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Reason: Convexity effecto The relation between bond prices and yield is not linear but

convex

truebond price followsthe blue curve line

durationis theslo e of the curve

badapproximation

Price

goo

approximation

e price using duration

follows the purple line



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, g g

Portfolio modified duration:o Weighted average of modified duration of the bonds in the

portfolio:

= ....2211 NNP

where wi = weight for bond i

i

N = number of bonds in the portfolio



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, g g

Examples:

annualcoupon,

$1milface

value)

(semi

annualcompoun

(Market

value)

duration

(years)

ding)

5.3%pa 2yr 6%pa $986,990 1.867 0.507

7.0%pa5yr 8%pa $959,446 4.122 0.493Total $1,946,436 1.000

ModDP 122.4493.0867.1507.0 +=

yrs979.2=

Interpretat on:T eport o omar etva uew

approximatelychangeby0.02979%fora1bp changeinthe

yield



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g g

Proof:o Price (market value) approximation for the bond portfolio for 1

bp change in yield:

)7(yPModDP PPP =

0001.0436,946,1979.2PP =

)(579$ termabsolutein=

%02979.0889,934,1

579==

PP

Por



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Proof using full valuation:

annualcoupon,

$1milface

value)

(semi

annualcompoun

(Market

value)

(increaseby

1bp)

price(market

value)

ding)

5.3%pa 2yr 6%pa $986,990 6.01%pa $986,806

7.0%pa5yr 8%pa $959,446 8.01%pa $959,050Total $1,946,436 $1,945,856

Difference=$580or0.02979%



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Important assumptions for bond portfolio modifiedduration:o Only provide a portfolio price approximation to a very small

change in the yieldso Assume a parallel shift in the term structure eg 6% 6.01%

AND 8% 8.01%



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Trading strategies using portfolio modified duration:o Longest portfolio modified duration provides maximum price

volatility (ie percentage price change)o

Hence, as an investor/asset manager: If you expect a decline in the yield, you should increase

por o o mo e ura on o max m ze on pr ceincrease. How to increase portfolio modified duration?

-and use the proceeds to long/buy long-term bonds orlong term futures

portfolio modified duration to minimize bond price decline. How to reduce portfolio modified duration?

sell/short long-term bonds or long-term futures anduse the proceeds to long/buy short-term bonds (egcommercial papers)

Duration, DV01 & hedgingInthetradinggame,youweretheissuerofa

portfolioofdebtratherthanabuyer(owner)ofa


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p y

Trading strategies using portfolio modified duration:o As an liability manager:

If you expect a decline in the yield, you should reduce

portfolio modified duration to have minimal bond priceincrease.ow o ecrease por o o mo e ura on uy-

back long-term bonds or long term futures andshort/sell/issue short-term bonds (eg commercial

If you expect an increase in the yield, you should increaseportfolio modified duration to have maximum bond price

How to increase portfolio modified duration?

sell/short/issue long-term bonds or long-term futuresand use the roceeds to buy-back short-term bonds(eg commercial papers)



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Hedging with modified duration:o A long position in bond B can be hedged by a short position in

bond Ho

Intuitive reason: If interest rate rises (ie bond price falls), welose in bond B (assuming we hold/long bond B), but we gain inon s nce we s or ssue on .

o A perfect hedge suggests the total DV01 of our portfolio iszero or equivalently, DV01B = DV01H.

0001.00001.0

HB

PodDPodD

PP

=

=

)8(H

B

B

H

PP

ModDModD =



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Examples:

annualcoupon,

$1mil

face

value)

annual

compounding)

(Market

value

for

1

bond)

duration

(years)

5.3%pa 2yr Long Unhedged

bondB

6%pa 986,990 1.867

7.0%pa5yr Short Hedgebond

H

8%pa $959,446 4.122

= BHP

P

odD

ModD

To erfectl hed eonebondBwitha

990,986867.1122.4 =

HP

marketvalueof$986,990,weneed

bondHwithamarketvalueof$447,043

ie 0.4659bondH

043,447=HP



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Proof using full valuation:o A 1 bp yield rise (6% 6.01%) decreases bond price B

(which we hold) from $986,990 to $986,806 ie $184

o A 1 bp yield rise (8% 8.01%) decreases bond price H(which we short) from $959,446 to $959,050 ie $395.

o Since we short 0.4659 bond H, we gain $184.03

o Total net portfolio value is about $0!

FINM7405 Interest Rate Risk 2011-W1

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Transcript of FINM7405 Interest Rate Risk 2011-W1