Chapter Sixteen
Managing Bond
Portfolios
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Chapter Overview
• Examine various fixed-income portfolio
strategies
• Distinguish between passive and active
approaches
• Discuss sensitivity of bond prices to interest
rates fluctuations
• Sensitivity is measured by duration
• Consider refinements in the way interest rate
sensitivity is measured, focusing on bond
convexity
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Interest Rate Sensitivity (1 of 2)
1. Bond prices and yields are inversely related
2. An increase in a bond’s yield to maturity
results in a smaller price change than a
decrease in yield of equal magnitude
3. Prices of long-term bonds tend to be more
sensitive to interest rate changes than prices
of short-term bonds
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Interest Rate Sensitivity (2 of 2)
4. Interest rate risk is less than proportional to
bond maturity
5. Interest rate risk is inversely related to the
bond’s coupon rate
6. The sensitivity of a bond’s price to a change
in its yield is inversely related to the YTM at
which the bond is currently selling
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Change in Bond Price as a Function
of Change in Yield to Maturity
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Prices of 8% Coupon Bond
(Coupons Paid Semiannually)
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Prices of Zero-Coupon Bond
(Semiannual Compounding)
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Duration
• A measure of the average maturity of a bond’s
promised cash flows
• Macaulay’s duration equals the weighted
average of the times to each coupon or principal
payment
• Weight applied to each payment time is proportion of
total value of bond accounted for by that payment
(i.e., the PV of the payment divided by the bond price)
• Duration = Maturity for zero coupon bonds
• Duration < Maturity for coupon bonds
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Duration Calculation
• Duration calculation:
T
D =  t wt
t =1
wt =
CFt
(1 + y )
t
P
CFt = Cash Flow at Time t
P = Price of Bond
y = Yield to Maturity
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Interest Rate Risk
• Duration as a measure of interest rate
sensitivity
• Price change is proportional to duration
  (1 + y ) 
P
= −D  

P
 1+ y 
• D* = Modified duration
P
= − D * y
P
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Duration Rules
(1 of 2)
• Rule 1
• The duration of a zero-coupon bond equals its
time to maturity
• Rule 2
• Holding maturity constant, a bond’s duration is
lower when the coupon rate is higher
• Rule 3
• Holding the coupon rate constant, a bond’s
duration generally increases with its time to
maturity
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Duration Rules
(2 of 2)
• Rule 4
• Holding other factors constant, the duration of a
coupon bond is higher when the bond’s yield to
maturity is lower
• Rule 5
• The duration of a level perpetuity is equal to:
1+ y
y
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Duration Rules
(2 of 2)
Example on Rule 5:
At a 10% yield, the duration of a perpetuity that pays
100$ once a year forever is 1.10/0.10= 11 years.
But at an 8% yield, 1.08/0.08= 13.5 years.
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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
(1 of 2)
• Relationship between bond prices and yields
is not linear
• Duration rule is a good approximation for only
small changes in bond yields
• Bonds with higher convexity exhibit higher
curvature in the price-yield relationship
• Convexity is measured as the rate of change of the
slope of the price-yield curve, expressed as a
fraction of the bond price
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Bond Price Convexity
(30-Year Maturity; 8% Coupon; Initial YTM = 8%)
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Convexity
(2 of 2)
𝑇
1
𝐶𝐹𝑡
2
𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 =
෍
(𝑡
+ 𝑡)
2
𝑡
𝑃 × (1 + 𝑦)
(1 + 𝑦)
𝑡=1
• Accounting for convexity changes the equation:
Δ𝑃
1
= −𝐷 ∗ Δ𝑦 + [Convexity × (Δ𝑦)2 ]
𝑃
2
Example 16.2 P(506)
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Convexity of Two Bonds
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Why Do Investors Like Convexity?
• Bonds with greater curvature gain more in
price when yields fall than they lose when
yields rise
• The more volatile interest rates, the more
attractive this asymmetry
• Investors must pay higher prices and accept
lower yields to maturity on bonds with greater
convexity
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Duration and Convexity of Callable
Bonds
• As rates fall, there is a ceiling on the bond’s
market price, which cannot rise above the call
price
• As rates fall, the bond is subject to price
compression
• Use effective duration (Change in
price/change in interest rates):
P P
Effective Duration =
r
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Price–Yield Curve for a
Callable Bond
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Duration and Convexity: MBS
• Mortgage-Backed Securities (MBS)
• Though the number of outstanding callable
corporate bonds has declined, the MBS market
has grown rapidly
• MBS are a portfolio of callable amortizing loans
• Homeowners may repay their loans at any time
• MBS have negative convexity
• Often sell for more than their principal balance
• Homeowners do not refinance as soon as rates drop, so
implicit call price is not a firm ceiling on MBS value
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Price-Yield Curve for a
Mortgage-Backed Security
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Passive Bond Management
• Passive managers take bond prices as fairly set
and seek to control only the risk of their fixedincome portfolio
• Two classes of passive management:
• Indexing strategy
• Immunization techniques
• Both classes accept market prices as being
correct, but differ greatly in terms of risk
exposure
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Bond-Index Funds
• Similar to stock market indexing
• Idea is to create a portfolio that mirrors the
composition of an index that measures the broad
market
• Challenges in construction:
• Very difficult to purchase each security in the index in
proportion to its market value
• Many bonds are very thinly traded
• Difficult rebalancing problems
• Due to challenges, a cellular approach is pursued
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Stratification of Bonds into Cells
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Passive Management: Immunization
• Immunization techniques are used to shield
overall financial status from interest rate risk
• Widely used by pension funds, insurers, and banks
• Duration-matched assets and liabilities let the
asset portfolio meet the firm’s obligations
despite interest rate movement
• Balances reinvestment rate risk and price risk
• Rebalancing is required to realign the portfolio’s
duration with the duration of the obligation
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Terminal Value of a 6-year Maturity
Bond Portfolio After 5 Years
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Growth of Invested Funds
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Immunization
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Cash Flow Matching and Dedication
• Cash flow matching is a form of immunization that
requires matching cash flows from a bond portfolio
with those of an obligation
• Imposes many constraints on bond selection process
• Cash flow matching in a multiperiod basis is referred
to as a dedication strategy
• Manager selects either zero-coupon of coupon bonds
with total cash flows in each period that match a series of
obligations
• Once-and-for-all approach to eliminating interest rate risk
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Active Bond Management:
Sources of Potential Profit
1. Substitution swap – exchange of one bond for
another more attractively priced bond with similar
attributes
2. Intermarket spread swap – switching from one
segment of the bond market to another (e.g., from
Treasuries to corporates)
3. Rate anticipation swap – switch made between
bonds of different durations in response to forecasts
of interest rates
4. Pure yield pickup swap – moving to higher-yield,
longer-term bonds to capture the liquidity premium
5. Tax swap – swapping two similar bonds to capture a
tax benefit
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Active Bond Management:
Horizon Analysis
• Horizon analysis involves forecasting the
realized compound yield over various holding
periods of investment horizons
• Analyst selects a particular holding periods and
predicts the yield curve at the end of the 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 its end-of-period price
calculated
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