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Investments, 8th edition
Bodie, Kane and Marcus
Slides by Susan Hine
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
CHAPTER 16 Managing Bond Portfolios
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• Inverse relationship between price and yield
• An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield
• Long-term bonds tend to be more price sensitive than short-term bonds
Bond Pricing Relationships
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16-3
Change in Bond Price as a Function of Change in Yield to Maturity
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• As maturity increases, price sensitivity increases at a decreasing rate
• Price sensitivity is inversely related to a bond’s coupon rate
• Price sensitivity is inversely related to the yield to maturity at which the bond is selling
Bond Pricing Relationships Continued
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16-5
Prices of 8% Coupon Bond (Coupons Paid Semiannually)
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Prices of Zero-Coupon Bond (Semiannually Compounding)
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• A measure of the effective maturity of a bond
• The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment
• Duration is shorter than maturity for all bonds except zero coupon bonds
• Duration is equal to maturity for zero coupon bonds
Duration
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t tt
w CF y ice ( )1 Pr
twtDT
t
1
CF Cash Flow for period tt
Duration: Calculation
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16-9
Calculating the Duration of Two Bonds
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Price change is proportional to duration and not to maturity
D* = modified duration
Duration/Price Relationship
(1 )
1
P yDx
P y
*P
D yP
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Rules for DurationRule 1 The duration of a zero-coupon bond
equals its time to maturity
Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is lower
Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity
Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower
Rules 5 The duration of a level perpetuity is equal to: (1+y) / y
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Bond Duration versus Bond Maturity
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Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)
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Convexity
• The relationship between bond prices and yields is not linear
• Duration rule is a good approximation for only small changes in bond yields
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Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8%
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Correction for Convexity
n
tt
t tty
CF
yPConvexity
1
22
)()1()1(
1
Correction for Convexity:
21 [ ( ) ]2P
D y Convexity yP
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Convexity of Two Bonds