Bond Risk Measure
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BOND RISK MEASUREBOND RISK MEASURE
Lada Kyj
November 19, 2003
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Bond Characteristics:Bond Characteristics:
Type of IssuerType of Issuer Governments (domestic, foreign, federal,
and municipal), government agencies, and
corporations can issue bonds.
Risk associated with bond.
Treasury bonds are regarded as most secure
and serve as a benchmark against which allbonds are compared.
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Bond Characteristics:Bond Characteristics:
MaturityMaturityMaturity denotes the date the bond will be
redeemed.
Term-to-maturity denotes the number of
years remaining until that date.
Indicates the number of coupon interest
payments the bond holder will receive.
A factor in determining the yield of a bond.
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Bond Characteristics:Bond Characteristics:
Coupon and PrincipalCoupon and Principal Principal(Face, Par) is the amount of the
debt.
Coupon rate is the rate of interest.
Coupon paymentis calculated as the
product of the coupon rate and the principal.
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Bond Characteristics:Bond Characteristics:
ProvisionsProvisions Provisions provide the issuer or holder the
right to retire debt prematurely or require
the issuer to retire a portion of outstanding
debt according to a specified schedule.
Examples: call provision, refunding
provision, sinking-fund provision, putprovisions, etc.
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Time Value of MoneyTime Value of Money Time value of money. A dollar today is
more valuable than a dollar in the future.
Present Value: 1/(1+r); where r is the
respective spot rate.
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Bond PricingBond PricingPrice is the sum of discounted cash flows.
Price = [c(t)/(1+rt)t
]- r is the spot rate, and c(t) is the payment at
time t.
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ExamplesExamples 1) Treasury zero expiring in a year, priced at $98.
98 = 100/(1+r1); r1= 0.0204
2) Treasury 4.25 expiring in 2 years, priced at $95.
95 = 4.25/(1+r1) + 104.25/(1+r2)2;
as r1= 0.0204, then r2 = 0.0713
OfNote: Can view coupon payments as a series ofzero coupon bonds.
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Law of One PriceLaw of One Price
The Law ofOnePrice dictates that two
securities with identical cash flows should
sell at the same price.
The spot rate extracted from one set of
bonds may be used to to price any set of
bonds with identical cash flows.
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Yield
Yield--toto--MaturityMaturity
Yield-to-Maturity is the single rate such that
discounting a securitys cash flow at that
rate produces the market price.
Price = [c(t)/(1+y)t]
Example:
Treasury 4.25 expiring in 2 years, priced at $95.
95 = 4.25/(1+y) + 104.25/(1+y)2; y = 0.0702
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PriceY
ield RelationshipPriceY
ield Relationship
yield
price
0.02 0.04 0.06 0.08 0.10 0.12 0.14
60
80
100
120
140
160
180
Price-yield
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Duration and ConvexityDuration and Convexity
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DurationDuration Duration is the measure of the approximate
sensitivity of a bonds value to rate changes.
Duration = -(1/P)(P/y)
Consider the first derivative of price divided
by price:(P/y)(1/P) = [(-t)c(t)(1+y)-t]/P(1/(1+y)) = D/(1+y)
The first derivative is the modified duration and D is
the Macaulay Duration.
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Types of DurationTypes of Duration1) MacaulayDuration weighted average number
of years until the bonds cash flows occur,
where the present values of each paymentrelative to the bonds price are used as weights.
2) ModifiedDuration Macaulay Duration dividedby (1 + yield). Assumes that changes in yield donot influence cash flows.
3) EffectiveDuration Recognition is given to thefact that yield changes may change the expectedcash flows.
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ConvexityConvexity Convexity measures how interest rate
sensitivity changes with rates.C = (d2P/dy2)(1/P) = [(t)(t+1)c(t)(1+y)-t]/P(1/(1+y) 2)
-A decline in yields creates stronger convexity
impacts than does an equivalent rise in yields.
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Second Order TaylorSecond Order Taylor
ApproximationApproximation Approximate price-yield function:
P(y+y) P(y) + (dP/dy)y + (1/2)(d2P/dy2)y2
Subtract P from both sides, and then divide by P:
P/P (1/P)(dP/dy)y + (1/2)(d2P/dy2)y2
= -Dy + (1/2)C y2
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Term Structure of InterestTerm Structure of Interest
RatesRates Term Structure ofInterest Rates measures
the relationship among the yields on
default-free securities that differ only intheir term to maturity.
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Expectations HypothesisExpectations Hypothesis Bonds are priced so that the implied forward rates
are equal to the expected spot rates. The only
reason for an upward-sloping term structure is thatthe investors expect future spot rates to be higher
than current spot rates. Fama (1984) tested the US
Treasury market from 1959 to 1982 and found that
the forward premium on average preceded a rise inthe spot rate, but less than would be predicted.
Lutz
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Liquidity PreferenceLiquidity Preference
HypothesisHypothesis Risk aversion will cause forward rates to be
systematically greater than expected spot
rates. The term premium is the incrementrequired to induce investors to hold longer-
term securities. Suggests that risk comes
solely from uncertainty about theunderlying real rate.
Hicks
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Market SegmentationMarket Segmentation
HypothesisHypothesis Postulates that individuals have strong
maturity preferences and that bonds of
different maturities trade in separate anddistinct markets. A shortcoming of this
hypothesis is that bonds of close maturities
will act as substitutes.Culbertson
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Preferred Habitat TheoryPreferred Habitat Theory States that the shape of the yield curve is
influenced by asset-liability management
constraints.
Modigliani and Sutch
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Curve Fitting:Curve Fitting:
Linear InterpolationLinear Interpolation Not differential at the
nodes
Poor approximation formissing values.
aturit
iel
3
3
5
Ter
tructure using linear interpolation
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Curve Fitting:Curve Fitting:
Piecewise CubicPiecewise Cubic
turit
iel
3
!
3
"
5
cubics are determined by 4parameters:
Y=ax3+bx2 +cx+d; there arem points, so m-1 intervals.
4(m-1) parameters
2(m-1)interpolationconditions, and m-2 first
derivative matchingconditions and m-2 secondorder matching condition,therefore two 2 natural
boundary conditions
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Curve Fitting:AdvancesCurve Fitting:AdvancesIoannides, M. 2003 A comparison of yield curve
estimation techniques using UK data. Journal of
Banking and Finance 27, 1-26.-Comparison is made by computing abnormal
returns in trading strategies.
-Splines are rejected in favor of
parsimonious functions.