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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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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
BuyBABi.e.lendcashtodrawer
Cansell(trade)BABtoanotherdiscounter Finalholder discounter receivesfacevalue
atmaturityofbill
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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
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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Shapes of interest rates term structure
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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Shapes of interest rates term structure
Flat curve, which usually leads to .
gure : apanese governmen y e curve on ov. , . ource: oom erg
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Shapes of interest rates term structure
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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Shapes of interest rates term structure
Humped curve
gure : . . reasury curve on ay , . ource: oom erg
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Shapes of interest rates term structure
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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Shapes of interest rates term structure
Inverted term structure signals economic recession. Proof:
16
183-month T-bill
10-year bond
8
10
12
14
rcentage
2
4
6P
0
Apr/53
Apr/56
Apr/59
Apr/62
Apr/65
Apr/68
Apr/71
Apr/74
Apr/77
Apr/80
Apr/83
Apr/86
Apr/89
Apr/92
Apr/95
Apr/98
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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Shapes of interest rates term structure
Inverted term structure signals economic recession. Proof:
4
5
1
2
ercentage
-2
-1
0
Apr/53
Apr/56
Apr/59
Apr/62
Apr/65
Apr/68
Apr/71
Apr/74
Apr/77
Apr/80
Apr/83
Apr/86
Apr/89
Apr/92
Apr/95
Apr/98
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
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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Fundamentals of bond pricing
Discrete compounding vs continuous compounding
y=4.8%p.a.
compoundedsemi
)3(1ln
+=
my
mr
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Fundamentals of bond pricing
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)
Cash flow (2) Discount factor (3) Present value (4) =
(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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Fundamentals of bond pricing
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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Advanced issues in bond pricing
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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Advanced issues in bond pricing
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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Advanced issues in bond pricing
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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Advanced issues in bond pricing
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)
Cash flow (2) Discount factor (3) Present value (4) =
(2) x (3)
0.5 26500 ? ?1.0 26500 ? ?
1.5 26500 ? ?
2.0 1026500 ? ?
Sum $987,334
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Advanced issues in bond pricing
In summaryo Pricing bond using spot rate and yield to maturity gives the same
result.
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Bond price volatility
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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Bond price volatility
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
Jan/70
Jan/72
Jan/74
Jan/76
Jan/78
Jan/80
Jan/82
Jan/84
Jan/86
Jan/88
Jan/90
Jan/92
Jan/94
Jan/96
Jan/98
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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Duration, DV01 & hedging
How to measure and manage interest rate?
Duration
DV01 Futures/forwardValueatRisk(VaR) wap
D ti DV01 & h d i
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Duration, DV01 & hedging
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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Duration, DV01 & hedging
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
Duration DV01 & hedging
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Duration, DV01 & hedging
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
Duration DV01 & hedging
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Duration, DV01 & hedging
Macaulay duration = 1.92 years
Modified duration (in years): )4(
1 +
=
my
ModD
867.106.0
9233.1=
=ModD
2
Duration DV01 & hedging
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Duration, DV01 & hedging
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%
Duration DV01 & hedging
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Duration, DV01 & hedging
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
Duration DV01 & hedging
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Duration, DV01 & hedging
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?
Duration, DV01 & hedging
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Duration, DV01 & hedging
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 $184
Duration, DV01 & hedging
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Duration, DV01 & hedging
Remember:o Modified duration only provides a price approximation to a
very small change in yield (eg 1 bp change in yield)
o Example: Use modified duration to calculate priceapproximation when the yield changes by 50 bp
)(9215$
0050.09869908673.1
valueabsolutein
P
=
=
o Full valuation:
annually)
5.50% $996,261 +$9270
. ,
6.50% $977,830 $9160
Duration, DV01 & hedging
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Duration, DV01 & hedging
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
Duration, DV01 & hedging
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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
Duration, DV01 & hedging
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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 $1,946,436 1.000
ModDP 122.4493.0867.1507.0 +=
yrs979.2=
Interpretat on:T eport o omar etva uew
approximatelychangeby0.02979%fora1bp changeinthe
yield
Duration, DV01 & hedging
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g g
Proof:o Price (market value) approximation for the bond portfolio for 1
bp change in yield:
)7(yPModDP PPP =
0001.0436,946,1979.2PP =
)(579$ termabsolutein=
%02979.0889,934,1
579==
PP
Por
Duration, DV01 & hedging
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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%
Duration, DV01 & hedging
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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%
Duration, DV01 & hedging
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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)
Duration, DV01 & hedging
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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 =
Duration, DV01 & hedging
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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
Duration, DV01 & hedging
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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!
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