Download - Fixed Income
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Zvi Wiener
Fixed Income
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Plan
• Pricing of Bonds
• Measuring yield
• Bond Price Volatility
• Factors Affecting Yields and the Term Structure of IR
• Treasury and Agency Securities Markets
• Corporate Debt Instruments
• Municipals
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Plan
• Non-US Bonds
• Mortgage Loans
• Mortgage Pass-Through Securities
• CMO and Stripped MBS
• ABS
• Bonds with Embedded Options
• Analysis of MBS
• Analysis of Convertible Bonds
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Plan
• Active Bond Portfolio Management
• Indexing
• Liability Funding Strategies
• Bond Performance Measurement
• Interest Rate Futures
• Interest Rate Options
• Interest Rate Swaps, Caps, Floors
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Characteristics of a Bond• Issuer
• Time to maturity
• Coupon rate, type and frequency
• Linkage
• Embedded options
• Indentures
• Guarantees or collateral
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Sources
• Fabozzi, “Bond Markets, Analysis and
Strategies”, Prentice Hall.
• P. Wilmott, Derivatives, Wiley.
• Hull, White, Manuscript.
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Sectors• Treasury sector: bills, notes, bonds
• Agency sector: debentures (no collateral)
• Municipal sector: tax exempt
• Corporate sector: US and Yankee issues– bonds, notes, structured notes, CP– investment grade and noninvestment grade
• Asset-backed securities sector
• MBS sector
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Basic terms• Principal• Coupon, discount and premium bonds• Zero coupon bonds• Floating rate bonds• Inverse floaters• Deferred coupon bonds• Amortization schedule• Convertible bonds
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Basic Terms
• The Money Market Account
• LIBOR = London Interbank Offer Rate, see BBA Internet site
• FRA = Forward Rate Agreement
• Repos, reverse repos
• Strips = Separate Trading of Registeres Interest and Principal of Securities
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Basic Terms
• gilts (bonds issued by the UK government)
• JGB = Japanese Government Bonds
• Yen denominated issued by non-Japanese institutions are called Samurai bonds
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Major risks
• Interest rate risk
• Default risk
• Reinvestment risk
• Currency risk
• Liquidity risk
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Time Value of Money
• present value PV = CFt/(1+r)t
• Future value FV = CFt(1+r)t
• Net present value NPV = sum of all PV-PV 5 5 5 5 105
5
4
1 )1(
105
)1(
5
rrPV
tt
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TT
T
tt
t
r
C
r
CPV
)1()1(1
Term structure of interest rates
TT
TT
tt
t
t
r
C
r
CPV
)1()1(1
Yield = IRR
TT
T
tt
t
y
C
y
Cice
)1()1(Pr
1
How do we know that there is a solution?
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Price-Yield Relationship
• Price and yield (of a straight bond) move in opposite directions.
yield
price
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General pricing formula
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1 )1()1()1()1(
nv
nn
ttv
t
rr
C
rr
CP
periodmonthssixindays
couponnextandsettlementbetweendaysv
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Accrued Interest
Accrued interest = interest due in full period*
(number of days since last coupon)/
(number of days in period between coupon payments)
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Day Count Convention
Actual/Actual - true number of days
30/360 - assume that there are 30 days in each month and 360 days in a year.
Actual/360
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Floater
The coupon rate of a floater is equal to a
reference rate plus a spread.
For example LIBOR + 50 bp.
Sometimes it has a cap or a floor.
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Inverse Floater
Is usually created from a fixed rate security.
Floater coupon = LIBOR + 1%
Inverse Floater coupon = 10% - LIBOR
Note that the sum is a fixed rate security.
If LIBOR>10% there is typically a floor.
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Price Quotes and Accrued Interest
Assume that the par value of a bond is $1,000.
Price quote is in % of par + accrued interest
the accrued interest must compensate the
seller for the next coupon.
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Annualizing Yield
Effective annual yield = (1+periodic rate)m-1 examples
Effective annual yield = 1.042-1=8.16%
Effective annual yield = 1.024-1=8.24%
price
ratecouponannualyieldcurrent
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Bond selling at Relationship
Par Coupon rate=current yield=YTM
Discount Coupon rate<current yield<YTM
Premium Coupon rate>current yield>YTM
Yield to call uses the first call as cashflow.
Yield of a portfolio is calculated with the total cashflow.
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YTM and Reinvestment Risk
• YTM assumes that all coupon (and
amortizing) payments will be invested at the
same yield.
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YTM and Reinvestment Risk• An investor has a 5 years horizon
Bond Coupon Maturity YTM
A 5% 3 9.0%
B 6% 20 8.6%
C 11% 15 9.2%
D 8% 5 8.0%
What is the best choice?
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Bond Price Volatility
Consider only IR as a risk factor
Longer TTM means higher volatility
Lower coupons means higher volatility
Floaters have a very low price volatility
Price is also affected by coupon payments
Price value of a Basis Point = price change resulting from a change of 0.01% in the yield.
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Duration and IR sensitivity
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Duration
nn y
M
y
C
y
C
y
CP
)1()1()1()1( 2
nn y
nM
y
nC
y
C
y
C
P
DurationMacaulay
)1()1()1(
2
)1(
112
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Duration
y
DurationMacaulayDurationModified
1
DurationModifiedPdy
dP 1
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DurationBond duration price impact of +1%
YTM
A 3 yr
B 1 yr
C 10 yr
D 20 yr
-3%-1%-10%-20%
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Measuring Price Change
errordydy
Pddy
dy
dPdP 2
2
2
)(2
1
P
errordy
ConvDdy
P
dP 2)(
2
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The Yield to Maturity
The yield to maturity of a fixed coupon bond y is given by
n
i
ytTi
iectp1
)()(
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Macaulay Duration
Definition of duration, assuming t=0.
p
ecTD
n
i
yTii
i
1
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Macaulay Duration
What is the duration of a zero coupon bond?
T
tt
tT
tt y
CFt
iceBondwtD
11 )1(Pr
1
A weighted sum of times to maturities of each coupon.
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Meaning of Duration
Dpecdy
d
dy
dp n
i
yTi
i
1
r
$
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Convexity
r
$
2
2
y
pC
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FRA Forward Rate AgreementA contract entered at t=0, where the parties (a lender
and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T].
Assuming continuous compounding we have
at time S: -K
at time T: KeR*(T-S)
Calculate the FRA rate R* which makes PV=0hint: it is equal to forward rate
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ALM Duration
• Does NOT work!• Wrong units of measurement• Division by a small number
r
A
ADA
1r
L
LDL
1
r
LA
LAD LA
)(1
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ALM Duration
A similar problem with measuring yield
r
P
VaR P 1
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Do not think of duration as a measure of time!
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• Key rate duration
• Principal component duration
• Partial duration
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Factors affecting Bond yields and TS
• Base interest rate - benchmark interest rate
• Risk Premium - spread
• Expected liquidity
• Market forces - Demand and supply
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Taxability of interest• qualified municipal bonds are exempts from
federal taxes.
After tax yield = pretax yield (1- marginal tax rate)
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Do not use yield curve to price bondsPeriod A B1-9 $6 $110 $106 $101They can not be priced by discounting
cashflow with the same yield because of different structure of CF.
Use spot rates (yield on zero-coupon Treasuries) instead!
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On-the-run Treasury issues
Off-the-run Treasury issues
Special securities
Lending
Repos and reverse repos
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Forward RatesBuy a two years bond
Buy a one year bond and then use the money to buy another bond (the price can be fixed today).
(1+r2)=(1+r1)(1+f12)
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Forward Rates
(1+r3)=(1+r1)(1+f13)= (1+r1)(1+f12)(1+f13)
Term structure of instantaneous forward rates.
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Determinants of the Term Structure
Expectation theory
Market segmentation theory
Liquidity theory
Mathematical models: Ho-Lee, Vasichek,
Hull-White, HJM, etc.
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• What is the duration of a floater?
• What is the duration of an inverse floater?
• How coupon payments affect duration?
• Why modified duration is better than
Macaulay duration?
• How duration can be used for hedging?
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