Investments, 8th edition

Bodie, Kane and Marcus

Slides by Susan HineSlides by Susan Hine

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

CHAPTER 16CHAPTER 16 Managing Bond Managing Bond PortfoliosPortfolios

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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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Figure 16.1 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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Table 16.1 Prices of 8% Coupon Bond (Coupons Paid Semiannually)

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Table 16.2 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 CashFlow for period tt

Duration: Calculation

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Spreadsheet 16.1 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 Duration

Rule 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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Figure 16.2 Bond Duration versus Bond Maturity

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Table 16.3 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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Figure 16.3 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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Figure 16.4 Convexity of Two Bonds

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Callable Bonds

• As rates fall, there is a ceiling on possible prices

– The bond cannot be worth more than its call price

• Negative convexity

• Use effective duration:/

Effective Duration = P P

r

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Figure 16.5 Price –Yield Curve for a Callable Bond

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Mortgage-Backed Securities

• Among the most successful examples of financial engineering

• Subject to negative convexity

• Often sell for more than their principal balance

– Homeowners do not refinance their loans as soon as interest rates drop

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Figure 16.6 Price -Yield Curve for a Mortgage-Backed Security

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Mortgage-Backed Securities Continued

• They have given rise to many derivatives including the CMO (collateralized mortgage obligation)

– Use of tranches

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Figure 16.7 Panel A: Cash Flows to Whole Mortgage Pool; Panels B–D Cash Flows to

Three Tranches

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• Bond-Index Funds

• Immunization of interest rate risk:

– Net worth immunizationDuration of assets = Duration of liabilities

– Target date immunizationHolding Period matches Duration

Passive Management

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Figure 16.8 Stratification of Bonds into Cells

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Table 16.4 Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested)

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Figure 16.9 Growth of Invested Funds

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Figure 16.10 Immunization

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Table 16.5 Market Value Balance Sheet

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Cash Flow Matching and Dedication

• Automatically immunize the portfolio from interest rate movement– Cash flow and obligation exactly offset each

other• i.e. Zero-coupon bond

• Not widely used because of constraints associated with bond choices

• Sometimes it simply is not possible to do

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• Substitution swap

• Intermarket swap

• Rate anticipation swap

• Pure yield pickup

• Tax swap

Active Management: Swapping Strategies

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Horizon Analysis

• Select a particular holding period and predict the yield curve at end of period

• Given a bond’s time to maturity at the end of the holding period

– Its yield can be read from the predicted yield curve and the end-of-period price can be calculated

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Contingent Immunization

• A combination of active and passive management

• The strategy involves active management with a floor rate of return

• As long as the rate earned exceeds the floor, the portfolio is actively managed

• Once the floor rate or trigger rate is reached, the portfolio is immunized

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Figure 16.11 Contingent Immunization