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Copyright 2005 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
10Bond Prices and Yields
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Bond Prices and Yields
Our goal in this chapter is to understand the relationshipbetween bond prices and yields.
In addition, we will examine some fundamental toolsthat fixed-income portfolio managers use when theyassess bond risk.
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Bond Basics, II.
Two basic yield measures for a bond are itscoupon rate and its current yield.
valuePar
couponAnnual
rateCoupon
priceBond
couponAnnual
yieldCurrent
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The Bond Pricing Formula
Recall: The price of a bond is found by adding together the
present value of the bonds coupon payments and the presentvalue of the bonds face value.
The formula is:
In the formula, C represents the annual coupon payments (in $),FV is the face value of the bond (in $), and M is the maturity of thebond, measured in years.
2M2M2
YTM1
FV
2YTM1
11
YTM
CPriceBond
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Example: Using the Bond Pricing Formula
What is the price of a straight bond with: $1,000 facevalue, coupon rate of 5%, YTM of 6%, and a maturity of10 years?
$925.61.
553.680.44632)(833.33
2
0.061
1000
2
0.061
11
0.06
50PriceBond
2YTM
1
FV
2YTM
1
11
YTM
CPriceBond
102102
2M2M
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Premium and Discount Bonds, I.
Bonds are given names according to the relationshipbetween the bonds selling price and its par value.
Premium bonds: price > par valueYTM < coupon rate
Discount bonds: price < par valueYTM > coupon rate
Par bonds: price = par valueYTM = coupon rate
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Premium and Discount Bonds, II.
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Relationships among Yield Measures
for premium bonds:coupon rate > current yield > YTM
for discount bonds:coupon rate < current yield < YTM
for par value bonds:coupon rate = current yield = YTM
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Calculating Yield to Maturity, I.
Suppose we know the current price of a bond, its coupon rate, andits time to maturity. How do we calculate the YTM?
We can use the straight bond formula, trying different yields until wecome across the one that produces the current price of the bond.
This is tedious. So, to speed up the calculation, financialcalculators and spreadsheets are often used.
52522
YTM1
$1,000
2YTM1
11
YTM
$90$1,083.17
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Yield to Call
Yield to call (YTC) is a yield measure that assumes a bond will be
called at its earliest possible call date.
The formula to price a callable bond is:
In the formula, C is the annual coupon (in $), CP is the call price ofthe bond, T is the time (in years) to the earliest possible call date,
and YTC is the yield to call, with semi-annual coupons.
As with straight bonds, we can solve for the YTC, if we know theprice of a callable bond.
2T2T2
YTC1
CP
2YTC1
11YTC
CPriceBondCallable
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Interest Rate Risk and Maturity
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Bond Prices and Yields
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Duration
Bondholders know that the price of their bonds change when
interest rates change. But, How big is this change?
How is this change in price estimated?
Macaulay Duration, or Duration, is the name of concept that helps
bondholders measure the sensitivity of a bond price to changes inbond yields. That is:
Two bonds with the same duration, but not necessarily the samematurity, will have approximately the same price sensitivity to a(small) change in bond yields.
2
YTM1
YTMinChangeDurationPriceBondinChangePct.
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Example: Using Duration
Example: Suppose a bond has a Macaulay Duration of 6 years,
and a current yield to maturity of 10%.
If the yield to maturity declines to 9.75%, what is the resultingpercentage change in the price of the bond?
-1.4286%
20.101
0.100.09756-PriceBondinChangePct.
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Modified Duration
Some analysts prefer to use a variation of Macaulays
Duration, known as Modified Duration.
The relationship between percentage changes in bondprices and changes in bond yields is approximately:
2
YTM1
DurationMacaulayDurationModified
YTMinChangeDurationModified-PriceBondinChangePct.
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Calculating Macaulays Duration
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Calculating Macaulays Duration
In general, for a bond paying constant semiannualcoupons, the formula for Macaulays Duration is:
In the formula, C is the annual coupon rate, M is thebond maturity (in years), and YTM is the yield tomaturity, assuming semiannual coupons.
1
2YTM1CYTM
YTMCM
2
YTM1
YTM 2
YTM1
Duration 2M
C l l ti M l D ti
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Calculating Macaulays Duration
for Par Bonds
If a bond is selling for par value, the duration formulacan be simplified to:
2M
2YTM1
11YTM
2YTM1DurationBondValuePar
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Properties of Duration
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Immunization by Duration Matching
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Useful Internet Sites
www.bondmarkets.com (Check out the bonds section)
www.bondsonline.com (Bond basics and currentmarket data)
www.jamesbaker.com (A practical view of bondportfolio management)
http://www.bondmarkets.com/http://www.bondsonline.com/http://www.jamesbaker.com/http://www.jamesbaker.com/http://www.bondsonline.com/http://www.bondmarkets.com/ -
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Chapter Review, I.
Bond Basics Straight Bonds
Coupon Rate and Current Yield
Straight Bond Prices and Yield to Maturity Straight Bond Prices
Premium and Discount Bonds
Relationships among Yield Measures
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Chapter Review, II.
More on Yields Calculating Yields
Yield to Call
Interest Rate Risk and Malkiels Theorems Promised Yield and Realized Yield
Interest Rate Risk and Maturity
Malkiels Theorems
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Chapter Review, III.
Duration Macaulay Duration
Modified Duration
Calculating Macaulays Duration
Properties of Duration
Dedicated Portfolios and Reinvestment Risk Dedicated Portfolios
Reinvestment Risk
Immunization
Price Risk versus Reinvestment Rate Risk Immunization by Duration Matching
Dynamic Immunization