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Chapter 16

Revision of the Fixed-Income Portfolio

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There are no permanent changes because change itself is permanent. It behooves the industrialist to

research and the investor to be vigilant.

- Ralph L. Woods

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Outline Introduction Passive versus active management

strategies Bond convexity

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Introduction Fixed-income security management is

largely a matter of altering the level of risk the portfolio faces:• Interest rate risk• Default risk• Reinvestment rate risk

Interest rate risk is measured by duration

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Passive Versus Active Management Strategies

Passive strategies Active strategies Risk of barbells and ladders Bullets versus barbells Swaps Forecasting interest rates Volunteering callable municipal bonds

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Passive Strategies Buy and hold Indexing

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Buy and Hold Bonds have a maturity date at which their

investment merit ceases

A passive bond strategy still requires the periodic replacement of bonds as they mature

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Indexing Indexing involves an attempt to replicate

the investment characteristics of a popular measure of the bond market

Examples are:• Salomon Brothers Corporate Bond Index• Lehman Brothers Long Treasury Bond Index

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Indexing (cont’d) The rationale for indexing is market

efficiency• Managers are unable to predict market

movements and that attempts to time the market are fruitless

A portfolio should be compared to an index of similar default and interest rate risk

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Active Strategies Laddered portfolio Barbell portfolio Other active strategies

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Laddered Portfolio In a laddered strategy, the fixed-income

dollars are distributed throughout the yield curve

A laddered strategy eliminates the need to estimate interest rate changes

For example, a $1 million portfolio invested in bond maturities from 1 to 25 years (see next slide)

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Laddered Portfolio (cont’d)

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Barbell Portfolio The barbell strategy differs from the

laddered strategy in that less amount is invested in the middle maturities

For example, a $1 million portfolio invests $70,000 par value in bonds with maturities of 1 to 5 and 21 to 25 years, and $20,000 par value in bonds with maturities of 6 to 20 years (see next slide)

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Barbell Portfolio (cont’d)

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Barbell Portfolio (cont’d) Managing a barbell portfolio is more complicated

than managing a laddered portfolio Each year, the manager must replace two sets of

bonds:• The one-year bonds mature and the proceeds are used

to buy 25-year bonds

• The 21-year bonds become 20-years bonds, and $50,000 par value are sold and applied to the purchase of $50,000 par value of 5-year bonds

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Other Active Strategies Identify bonds that are likely to experience

a rating change in the near future• An increase in bond rating pushes the price up

• A downgrade pushes the price down

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Risk of Barbells and Ladders Interest rate risk Reinvestment rate risk Reconciling interest rate and reinvestment

rate risks

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Interest Rate Risk Duration increases as maturity increases

The increase in duration is not linear• Malkiel’s theorem about the decreasing

importance of lengthening maturity• E.g., the difference in duration between 2- and

1-year bonds is greater than the difference in duration between 25- and 24-year bonds

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Interest Rate Risk (cont’d) Declining interest rates favor a laddered

strategy

Increasing interest rates favor a barbell strategy

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Reinvestment Rate Risk The barbell portfolio requires a reinvestment each

year of $70,000 par value The laddered portfolio requires the reinvestment

each year of $40,000 par value

Declining interest rates favor the laddered strategy Rising interest rates favor the barbell strategy

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Reconciling Interest Rate & Reinvestment Rate Risks

The general risk comparison:

Rising Interest Rates Falling Interest Rates

Interest Rate Risk Barbell favored Laddered favored

Reinvestment Rate Risk Barbell favored Laddered favored

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Reconciling Interest Rate & Reinvestment Rate Risks

The relationships between risk and strategy are not always applicable:• It is possible to construct a barbell portfolio

with a longer duration than a laddered portfolio– E.g., include all zero-coupon bonds in the barbell

portfolio

• When the yield curve is inverting, its shifts are not parallel

– A barbell strategy is safer than a laddered strategy

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Bullets Versus Barbells A bullet strategy is one in which the bond

maturities cluster around one particular maturity on the yield curve

It is possible to construct bullet and barbell portfolios with the same durations but with different interest rate risks• Duration only works when yield curve shifts are

parallel

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Bullets Versus Barbells (cont’d)

A heuristic on the performance of bullets and barbells:• A barbell strategy will outperform a bullet

strategy when the yield curve flattens

• A bullet strategy will outperform a barbell strategy when the yield curve steepens

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Swaps Purpose Substitution swap Intermarket or yield spread swap Bond-rating swap Rate anticipation swap

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Purpose In a bond swap, a portfolio manager

exchanges an existing bond or set of bonds for a different issue

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Purpose (cont’d) Bond swaps are intended to:

• Increase current income• Increase yield to maturity• Improve the potential for price appreciation

with a decline in interest rates• Establish losses to offset capital gains or

taxable income

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Substitution Swap In a substitution swap, the investor

exchanges one bond for another of similar risk and maturity to increase the current yield• E.g., selling an 8% coupon for par and buying

an 8% coupon for $980 increases the current yield by 16 basis points

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Substitution Swap (cont’d) Profitable substitution swaps are

inconsistent with market efficiency

Obvious opportunities for substitution swaps are rare

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Intermarket or Yield Spread Swap

The intermarket or yield spread swap involves bonds that trade in different markets• E.g., government versus corporate bonds

Small differences in different markets can cause similar bonds to behave differently in response to changing market conditions

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Intermarket or Yield Spread Swap (cont’d)

In a flight to quality, investors become less willing to hold risky bonds• As investors buy safe bonds and sell more risky

bonds, the spread between their yields widens Flight to quality can be measured using the

confidence index• The ratio of the yield on AAA bonds to the

yield on BBB bonds

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Bond-Rating Swap A bond-rating swap is really a form of

intermarket swap

If an investor anticipates a change in the yield spread, he can swap bonds with different ratings to produce a capital gain with a minimal increase in risk

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Rate Anticipation Swap In a rate anticipation swap, the investor

swaps bonds with different interest rate risks in anticipation of interest rate changes• Interest rate decline: swap long-term premium

bonds for discount bonds

• Interest rate increase: swap discount bonds for premium bonds or long-term bonds for short-term bonds

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Forecasting Interest Rates Few professional managers are consistently

successful in predicting interest rate changes

Managers who forecast interest rate changes correctly can benefit• E.g., increase the duration of a bond portfolio is

a decrease in interest rates is expected

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Volunteering Callable Municipal Bonds

Callable bonds are often retied at par as part of the sinking fund provision

If the bond issue sells in the marketplace below par, it is possible:• To generate capital gains for the client • If the bonds are offered to the municipality

below par but above the market price

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Bond Convexity The importance of convexity Calculating convexity General rules of convexity Using convexity

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The Importance of Convexity Convexity is the difference between the

actual price change in a bond and that predicted by the duration statistic

In practice, the effects of convexity are minor

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The Importance of Convexity (cont’d)

The first derivative of price with respect to yield is negative• Downward sloping curves

The second derivative of price with respect to yield is positive• The decline in bond price as yield increases is

decelerating• The sharper the curve, the greater the convexity

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The Importance of Convexity (cont’d)

Greater Convexity

Yield to Maturity

Bon

d P

rice

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The Importance of Convexity (cont’d)

As a bond’s yield moves up or down, there is a divergence from the actual price change (curved line) and the duration-predicted price change (tangent line)• The more pronounced the curve, the greater the

price difference

• The greater the yield change, the more important convexity becomes

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The Importance of Convexity (cont’d)

Yield to Maturity

Bon

d P

rice

Error from using duration only

Current bond price

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Calculating Convexity The percentage change in a bond’s price

associated with a change in the bond’s yield to maturity:

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2

1 1 Error( )

2

where bond price

yield to maturity

dP dP d PdR dR

P P dR P dR P

P

R

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Calculating Convexity (cont’d) The second term contains the bond

convexity:

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2

1Convexity ( )

2

d PdR

P dR

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Calculating Convexity (cont’d) Modified duration is related to the

percentage change in the price of a bond for a given change in the bond’s yield to maturity• The percentage change in the bond price is

equal to the negative of modified duration multiplied by the change in yield

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Calculating Convexity (cont’d) Modified duration is calculated as follows:

Macaulay duration

Modified duration1 Annual yield to maturity / 2

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General Rules of Convexity There are two general rules of convexity:

• The higher the yield to maturity, the lower the convexity, everything else being equal

• The lower the coupon, the greater the convexity, everything else being equal

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Using Convexity Given a choice, portfolio managers should

seek higher convexity while meeting the other constraints in their bond portfolios• They minimize the adverse effects of interest

rate volatility for a given portfolio duration