14 Fixed Income Portfolio Management

Chapter 14Fixed Income Portfolio Management

Business 4179

Important Terms

Barbell strategy Bond swap Bullet strategy Confidence index convexity

Flight to quality Laddered strategy Macaulay duration Modified duration Yield curve

inversion

Definitions – Confidence Index

Confidence Index is the ratio of the yield on AAA bonds to the yield on BBB bonds.

It has an upper boundary of 1.0 because the yield on safe bonds should never exceed the yield on risky bonds.

It is a measure of yield spread…as spreads widen, this indicates ‘smart money’ is becoming increasingly pessimistic about the future. The confidence index will fall.

Some equity analysts use this index to forecast trends in the equity markets based on the assumption that it takes longer for equity markets to respond to new expectations. They base this upon the assumption that equity markets have larger numbers of novice/inexperienced investors…bond markets are dominated by institutional money and portfolio managers.

8846.%2.5

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BBB

AAA

BBB

AAA

Yield

YieldCI

Yield

YieldCI

Yield Curves

Term to Maturity In Years

YTM %

Yield curve for BBB corporates

Yield curve for AAA corporates

Yield Spread

Definitions - Duration

Is the first derivative of the bond-pricing equation with respect to yield.

Types of Duration

Macaulay duration – a measure of time flow of cash from a bond.

Modified duration – a slight modification of Macaulay’s to account for semi-annual coupon payments

Effective duration – a direct measure of interest rate sensitivity of a bond price

Empirical duration – measures directly the percentage price change of a bond for an actual change in interest rates.

Definitions – Modified Duration

Is Macaulay’s duration adjusted for semi-annual coupon payments:

)]2

Maturity toYield Annual([1

durationMacaulay Duration Modified

Duration

An alternative measure of bond price sensitivity is the bond’s duration.

Duration measures the life of the bond on a present value basis.

Duration can also be thought of as the average time to receipt of the bond’s cashflows.

The longer the bond’s duration, the greater is its sensitivity to interest rate changes.

Duration and Coupon Rates

A bond’s duration is affected by the size of the coupon rate offered by the bond.

The duration of a zero coupon bond is equal to the bond’s term to maturity. Therefore, the longest durations are found in stripped bonds or zero coupon bonds. These are bonds with the greatest interest rate elasticity.

The higher the coupon rate, the shorter the bond’s duration. Hence the greater the coupon rate, the shorter the duration, and the lower the interest rate elasticity of the bond price.

Duration

The numerator of the duration formula represents the present value of future payments, weighted by the time interval until the payments occur. The longer the intervals until payments are made, the larger will be the numerator, and the larger will be the duration. The denominator represents the discounted future cash flows resulting from the bond, which is the bonds present value.

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Duration Example

As an example, the duration of a bond with $1,000 par value and a 7 percent coupon rate, three years remaining to maturity, and a 9 percent yield to maturity is:

years

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years 2.8 duration Modified

Duration Example

As an example, the duration of a bond with $1,000 par value and a 7 percent coupon rate, three years remaining to maturity, and a 9 percent yield to maturity is:

Yield to Maturity = 0.09

Time Cashflow PVIFPresent Value

Relative Weights

Time Weighted Present Value

Relative Weights

1 70 0.917431 64.22018349 0.067645 0.06764482 70 0.84168 58.91759953 0.062059 0.12411883 1070 0.772183 826.2363237 0.870296 2.6108875

Bond Price = 949.3741067 Duration = 2.8026511 years

Modified Duration = 2.6819627 years

Duration Example ...

As an example, the duration of a zero-coupon bond with $1,000 par value and three years remaining to maturity, and a 9 percent yield to maturity is:

years

DUR

0.3

)09.1(

1000$)09.1(

)3(1000$

3

3

23

Duration

is a handy tool because it can encapsule interest rate exposure in a single number.

rather than focus on the formula...think of the duration calculation as a process...

semi-annual duration calculations simply call for halving the annual coupon payments and discounting every 6 months.

24

Duration Rules-of-Thumb

duration of zero-coupon bond (strip bond) = the term left until maturity.

duration of a consol bond (ie. a perpetual bond) = 1 + (1/R)

where: R = required yield to maturity duration of an FRN (floating rate note) = 1/2 year

25

Other Duration Rules-of-Thumb

Duration and Maturity duration increases with maturity of a fixed-income asset, but at a

decreasing rate.

Duration and Yield duration decreases as yield increases.

Duration and Coupon Interest the higher the coupon or promised interest payment on the

security, the lower its duration.

26

Economic Meaning of Duration

duration is a direct measure of the interest rate sensitivity or elasticity of an asset or liability. (ie. what impact will a change in YTM have on the price of the particular fixed-income security?)

interest rate sensitivity is equal to:

dP = - D [ dR/(1+R)]

P

Where: P = Price of bond

C = Coupon (annual)

R = YTM

N = Number of periods

F = Face value of bond

Problems with Duration

It assumes a straight line relationship between the changes in bond price given the change in yield to maturity…however, the actual relationship is curvilinear…therefore, the greater the change in YTM, the greater the error in predicted bond price using duration…as can be seen…

The Problem with Duration

Bond Price

Yield to Maturity (%)

$1,200

$1,100

$1,000

$900

Current bond price

Actual Bond Price

Bond Price predicted by

duration

Error from using duration

only

Uses of Duration

Immunization strategies: If you equate the duration of an asset (bond)

with the duration of a liability, you will (subject to some limitations) immunize your investment portfolio from interest rate risk.

Used in predicting bond prices given a change in interest rates (yields)

Predicting a Bond Price using Duration

Price movements of bonds will vary proportionally with modified duration for small changes in yields.

An estimate of the percentage change in bond price equals the change in yield times the modified duration.

100.by divided points basisin change yield thei

bond theofduration modified theD-

bond for the price beginning theP

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100

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mod

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Predicting a Bond Price Using Duration



Relative Weights

Time Weighted Present Value

Relative Weights

1 70 0.917431 64.22018349 0.06764476 0.0676452 70 0.84168 58.91759953 0.06205941 0.1241193 1070 0.772183 826.2363237 0.87029583 2.610887

Bond Price = 949.3741067 Duration = 2.802651 years

Modified Duration = 2.681963 yearsPredicted change in interest rates (basis points) = 0.5Predicted percentage change in bond price=-D(mod) times 0.5 =-1.340981Predicted bond price = 936.6431768

Actual Bond Price



1 70 0.913242 63.926940642 70 0.834011 58.380767713 1070 0.761654 814.969621

Bond Price = 937.2773293

As you can see from the previous slide, using modified duration and predicting an increase of 0.5% in yield, we forecast the bond price to be $936.64 where as it will actually be $937.27.

Definitions - Convexity

Bond convexity is the difference between the actual price change of a bond and that predicted by the duration statistic.

It is the second derivative of the bond-pricing equation with respect to yield.

The importance of convexity increases as the magnitude of rate changes increases.

Other rules: 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. The greatest convexity would be observed for stripped bonds

at low yields.

Second Derivative of the Bond Pricing Equation

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Convexity

Bond Price

Yield to Maturity (%)

$1,200

$1,100

$1,000

$900

Notice the slope is less as the

yield increases.

Computation of Convexity

3-year bond, 12% coupon, 9% YTM

Yield to Maturity = 0.09(1) (2) "(3) (4) (5) (6)


t squared plus t =(4) times (5)

1 120 0.917431 110.0917431 2 220.18348622 120 0.84168 101.0015992 6 606.00959523 1120 0.772183 864.8454977 12 10378.14597

Bond Price = $1,075.94 $11,204.34 years

75.81,075.95

9,411.78Convexity

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Bond Portfolio Strategy

Selection of the most appropriate strategy involves picking one that is consistent with the objectives and policy guidelines of the client or institution.

There are two basic types of strategies:• Active

Laddered Barbell Bullet

Swaps (substitution, inter-market or yield spread, bond-rating, rate anticipation)

• Passive buy and hold, and indexing

Active Bond Portfolio Strategies Other authors categorize bond strategies as follows (see Frank K.

Reilly and Keith C. Brown, Investment Analysis and Portfolio Management.) Passive Portfolio Strategies

• Buy and hold• Indexing

Active Management Strategies• Interest rate anticipation• Valuation analysis• Credit analysis• Yield spread analysis• Bond swaps

Matched-funding strategies• Dedicated portfolio – exact cash match• Dedicated portfolio – optimal cash match and reinvestment• Classical (“pure”) immunization• Horizon matching

Contingent procedures (structured active management)• Contingent immunization• Other contingent procedures

Buy-and-Hold Strategy

Involves:• Finding issues with desired quality, coupon levels, term to

maturity, and important indenture provisions, such as call features

• Looking for vehicles whose maturities (or duration) approximate their stipulated investment horizon to reduce price and reinvestment rate risk.

A modified buy and hold strategy involves:• Investing with the intention of holding until maturity,

however, they still actively look for opportunities to trade into more desirable positions.

Indexing Strategy

The manager builds a portfolio that will match the performance of a selected bond-market index such as the Lehman Brothers Index, Scotia McLeod bond index, etc.• In such a case, the bond manager is NOT judged on the basis of

risk and return compared to an index, BUT on how closely the portfolio tracks the index.

Tracking error equals the difference between the rate of return for the portfolio and the rate of return for the bond-market index.

When a portfolio has a return of 8.2 percent and the index an 8.3 percent return, the tracking error would be 10 basis points.

Active Portfolio Strategies

There are three sources of return from holding a fixed-income portfolio:• coupon income• any capital gain (or loss),• reinvestment income

in general, the following factors affect a portfolio’s return:• changes in the level of interest rates• changes in the shape of the yield curve• changes in the yield spreads among bond sectors• changes in the yield spread (risk premium) for a particular bond (perhaps

the default risk associated with a particular bond increases or decreases)

Manager Expectations vs. Market Consensus

A money manager who pursues an active strategy will position a portfolio to capitalize on expectations about future interest rates.

But the potential outcome (as measured by total return) must be assessed before an active strategy is implemented.• The primary reason for assessing the potential outcome is that

the market (collectively) has certain expectations for future interest rates, and these expectations are embodied in the market price of bonds.

Yield Curve Strategies

Yield curve strategies involve positioning a portfolio to capitalize on expected changes in the shape of the yield curve.

A shift in the yield curve refers to the relative change in the yield for each Treasury maturity.• A parallel shift in the yield curve refers to a shift in which the

change in the yield on all maturities is the same.• A nonparallel shift in the yield curve means that the yield for each

maturity does not change by the same number of basis points. Historically, two types of nonparallel yield curve shifts have been

observed: a twist in the slop of the yield curve and a change in the humpedness of the yield curve.

Upward Parallel ShiftY

ield

cha

nge

0Short Intermediate Long

maturity

Downward Parallel ShiftY

ield

cha

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maturity

Nonparallel Shifts

A nonparallel shift in the yield curve means that the yield for each maturity does not change by the same number of basis points.

• Historically, two types of nonparallel yield curve shifts have been observed: a twist in the slope of the yield curve and a change in the humpedness of the yield curve.

Yield Curve Shifts

A flattening of the yield curve means that the yield spread between the yield on a long-term and a short-term Treasury has decreased.

A steepening of the yield curve means that the yield spread between a long-term and a short-term Treasury has increased.

Flattening TwistY

ield

cha

nge


maturity

Steepening TwistY

ield

cha

nge


maturity

Non-parallel Yield Curve Shifts

A change in the humpedness of the yield curve is referred to as a butterfly shift.

This is also an example of a non-parallel shift.

Positive ButterflyY

ield

cha

nge


maturity

Negative ButterflyY

ield

cha

nge


maturity

Yield Curve Shifts 1979-1990

Frank Jones found that the three types of yield curve shifts are NOT independent. (parallel, twists and butterfly)

The two most common shifts:• a downward shift combined with a

steepening of the yield curve, and• an upward shift combined with a flattening of

the yield curve.

Upward shift/flattening/positive butterfly

Flattening

Yie

ld

Parallel

Positivebutterfly

Term to Maturity

Downward shift/steepening/negative butterfly

Yie

ld

Parallel

Steepening

Negativebutterfly

Term to Maturity

Yield Curve Shifts and returns

Jones found that parallel shifts and twists in the yield curve are responsible for 91.6% of Treasury returns, while 3.4% of the returns is attributable to butterfly shifts, and the balance, 5%, to unexplained factor shifts.

This indicates that yield curve strategies require a forecast of

the direction of the shift and a forecast of the type of twist.


In portfolio strategies that seek to capitalize on expectations based on short-term movements in yields, the dominant source of return is the change in the price of the securities of the portfolio.

This means that the maturity of the securities in the portfolio will have an impact on the portfolio’s return……

• a total return over a 1-year investment horizon for a portfolio consisting of securities all maturing in 1 year will not be sensitive to changes in how the yield curve shifts 1 year from now.

• In contrast, the total return over a 1-year investment horizon for a

portfolio consisting of securities all maturing in 30 years will be sensitive to how the yield curve shifts because, 1 year from now, the value of the portfolio will depend on the yield offered on 20-year securities.

Yield Curve Strategies(bullet, barbell, and ladder)

Bullet StrategySpikes indicate maturing principalComment: Bullet concentrated around year 10

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Barbell StrategySpikes indicate maturing principalComment: Barbell below and

above 10 years

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Ladder strategySpikes indicate maturing principalComment: Laddered up to year 20


Each of these strategies (bullet, barbell, ladder) will result in different performance when the yield curve shifts.

The actual performance will depend on both the type of shift and the magnitude of the shift….thus, no general statements can be made about the optimal yield curve strategy.

Duration and Yield Curve Shifts

Duration is a measure of the sensitivity of the price of a bond or the value of a bond portfolio to changes in market yields.• A portfolio with a duration of 4 means that if market yields

increase by 100 basis points, the portfolio will change by approximately 4%.

• If a portfolio of bonds is made up of 5-year, 10-year and 20-year bonds, and the portfolio’s duration is 4…the portfolio’s value will change by 4% if the yields on all bonds change by 100 basis points. That is, it is assumed that there is a parallel yield curve shift.

Analysis of Expected Yield Curve Strategies

The proper way to analyze any portfolio strategy is to look at its potential total return.

Example:• consider the following two yield curve

strategies: Bullet portfolio: 100% bond C Barbell portfolio: 50.2% bond A and 49.8%

bond B

Three Hypothetical Treasury Securities

Bond Coupon Maturity(years)

Price PlusAccruedInterest

Yield toMaturity

DollarDuration

DollarConvexity

A 8.50% 5 100 8.50% 4.005 19.8164

B 9.50 20 100 9.50% 8.882 124.1702

C 9.25 10 100 9.25% 6.434 55.4506

•The bullet portfolio consists of only bond C, the 10-year bond. All principal is received when bond C matures in 10 years.

•The barbell portfolio consists of almost equal amount of the short-term and long-term securities. The principal will be received at two ends of the maturity spectrum. (5 year and 20 year dates).

Barbell and Bullet Duration

Bond Coupon Maturity(years)

Price PlusAccruedInterest

Yield toMaturity

DollarDuration

DollarConvexity

A 8.50% 5 100 8.50% 4.005 19.8164

B 9.50 20 100 9.50% 8.882 124.1702

C 9.25 10 100 9.25% 6.434 55.4506

•The dollar duration of the bullet portfolio per 100-basis-point change in yield is 6.43409.

•Dollar duration is a measure of the dollar price sensitivity of a bond or a portfolio.

•The dollar duration for the barbell portfolio is just the weighted average of the dollar duration of the two bonds:

= .502(4.005) + 0.498(8.882) = 6.434

Dollar Convexity

• Duration is just a first approximation• Convexity is the second derivative and simply gives a more accurate

indication of sensitivity of bond price to a change in interest rates.• Both duration and convexity assume a parallel shift in the yield

curve!! • Two general rules for 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.

• Managers should seek high convexity while meeting other constraints in their bond portfolios…by doing so, they minimize the adverse effects of interest rate volatility for a given portfolio duration.

Swaps

Swaps are used to do one of four things:• 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.

Substitution Swap

• Purpose- to increase current yield• Assumes market inefficiency…that

results in equally risky bonds (default risk, same duration) to have different prices…this is an arbitrage action

• In an efficient market, we expect few of these situations to arise.

Intermarket or Yield Spread Swap

Purpose- to take advantage of expected changes in the default risk premiums that may occur as a result of changes in market optimism or pessimism.

A confidence index measures these changes.

Bond-Rating Swap

Purpose- to take advantage of expected changes in the default risk premiums that may occur as a result of changes in bond ratings.

Fundamental analysis of the prospectus of the individual issuer and of their financial health is used to predict changes in bond ratings (but this must be done in conjunction with analysis of changes in the overall market returns (ie. yield curve changes).)

Rate Anticipation Swap

The purpose is to take advantage of expected changes in interest rates by positioning the bond portfolio with an appropriate duration, AND an appropriate default risk category.

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14 Fixed Income Portfolio Management