Chapter Overview
• Examine various fixed-income portfolio
Chapter Sixteen strategies
• Distinguish between passive and active
approaches
Managing Bond • Discuss sensitivity of bond prices to interest
rates fluctuations
Portfolios • 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) Interest Rate Sensitivity (2 of 2)
1. Bond prices and yields are inversely related 4. Interest rate risk is less than proportional to
bond maturity
• As maturity increases, price sensitivity increases at a
2. An increase in a bond’s yield to maturity decreasing rate
results in a smaller price change than a
decrease in yield of equal magnitude 5. Interest rate risk is inversely related to the
bond’s coupon rate
3. Prices of long-term bonds tend to be more 6. The sensitivity of a bond’s price to a change in
sensitive to interest rate changes than prices its yield is inversely related to the YTM at which
of short-term bonds the bond is currently selling
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Change in Bond Price as a Function Prices of 8% Coupon Bond
of Change in Yield to Maturity (Coupons Paid Semiannually)
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Prices of Zero-Coupon Bond Duration
(Semiannual Compounding)
• 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 Interest Rate Risk
• Duration calculation: • Duration as a measure of interest rate sensitivity
T • Price change is proportional to duration
D t wt P 1 y
D
1 y t 1
t
wt
CFt P 1 y
P • D* = Modified duration (= Mac D / (1+y) )
CFt Cash Flow at Time t P
P Price of Bond D * y
P
y Yield to Maturity
• Semi-annual compounding with YTM y:
• (simplified version; in practice analysts use time factors) Mod D = Mac D / (1 + y /2)
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Duration Rules Duration Rules
(1 of 2) (2 of 2)
• Rule 1 • Rule 4
• The duration of a zero-coupon bond equals its • Holding other factors constant, the duration of a
time to maturity coupon bond is higher when the bond’s yield to
• Rule 2 maturity is lower
• Holding maturity constant, a bond’s duration is • Rule 5
lower when the coupon rate is higher
• The duration of a level perpetuity is equal to:
• Rule 3 1 y
• Holding the coupon rate constant, a bond’s y
duration generally increases with its time to
maturity • Price of level perpetuity = C / y
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Bond Durations
Bond Duration versus Bond Maturity
(Yield to Maturity = 8% APR; Semiannual Coupons)
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Convexity Bond Price Convexity
(1 of 2) (30-Year Maturity; 8% Coupon; Initial YTM = 8%)
• 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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Convexity Convexity of Two Bonds
(2 of 2)
𝑇
1 𝐶𝐹𝑡
𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 = (𝑡 2 + 𝑡)
𝑃 × (1 + 𝑦)2 (1 + 𝑦)𝑡
𝑡=1
• Accounting for convexity changes the equation:
Δ𝑃 1
= −𝐷 ∗ Δ𝑦 + [Convexity × (Δ𝑦)2 ]
𝑃 2
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Duration and Convexity of Callable
Why Do Investors Like Convexity?
Bonds
• Bonds with greater curvature gain more in • As rates fall, there is a ceiling on the bond’s
price when yields fall than they lose when market price, which cannot rise above the call
yields rise price
• The more volatile interest rates, the more • As rates fall, the bond is subject to price
attractive this asymmetry compression
• Use effective duration:
• Investors must pay higher prices and accept
P P
lower yields to maturity on bonds with greater Effective Duration
convexity r
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Price–Yield Curve for a
Duration and Convexity: MBS
Callable Bond
• 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
Duration and Convexity: CMO
Mortgage-Backed Security
• Collateralized Mortgage Obligation (CMO)
• Further redirects the cash flow stream of the MBS
to several classes of derivative securities called
“tranches”
• Tranches may be designed to allocate interest rate
risk to investors most willing to bear that risk
• Different tranches may receive different coupon rates
• Some may be given preferential treatment in terms of
uncertainty over mortgage prepayment speeds
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Passive Bond Management Bond-Index Funds
• Passive managers take bond prices as fairly set • Similar to stock market indexing
and seek to control only the risk of their fixed- • Idea is to create a portfolio that mirrors the
income portfolio composition of an index that measures the broad
market
• Two classes of passive management:
• Challenges in construction:
• Indexing strategy
• Very difficult to purchase each security in the index in
• Immunization techniques proportion to its market value
• Both classes accept market prices as being • Many bonds are very thinly traded
correct, but differ greatly in terms of risk • Difficult rebalancing problems
exposure • Due to challenges, a cellular approach is pursued
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Stratification of Bonds into Cells 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
• Value of assets match liabilities whether rates rise/fall
• 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
Growth of Invested Funds
Bond Portfolio After 5 Years
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Market Value Balance Sheet Immunization
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Active Bond Management:
Cash Flow Matching and Dedication
Sources of Potential Profit
• Cash flow matching is a form of immunization that 1. Substitution swap – exchange of one bond for
another more attractively priced bond with similar
requires matching cash flows from a bond portfolio attributes
with those of an obligation 2. Intermarket spread swap – switching from one
• Imposes many constraints on bond selection process segment of the bond market to another (e.g., from
Treasuries to corporates)
• Cash flow matching on a multiperiod basis is 3. Rate anticipation swap – switch made between
referred to as a dedication strategy bonds of different durations in response to forecasts
of interest rates
• Manager selects either zero-coupon or coupon bonds
4. Pure yield pickup swap – moving to higher-yield,
with total cash flows in each period that match a series of longer-term bonds to capture the liquidity premium
obligations
5. Tax swap – swapping two similar bonds to capture a
• Once-and-for-all approach to eliminating interest rate risk 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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