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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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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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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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.

Bond Risk Measure

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