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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