Chapter 16
Revision of the
Fixed-Income Portfolio
1
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
2
Outline
 Introduction
 Passive
versus active management
strategies
 Bond convexity
3
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
4
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
5
Passive Strategies
 Buy
and hold
 Indexing
6
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
7
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
8
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
9
Active Strategies
 Laddered
portfolio
 Barbell portfolio
 Other active strategies
10
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 $1 million portfolio invested
in bond maturities from 1 to 25 years (see
next slide)
11
Par Value Held ($ in Thousands)
Laddered Portfolio (cont’d)
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
1
3
5
7
9
11
13
15
17
Years Until Maturity
19
21
23
25
12
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 $20,000
par value in bonds with maturities of 6 to 20
years (see next slide)
13
Par Value Held ($ in Thousands)
Barbell Portfolio (cont’d)
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
1
3
5
7
9
11
13
15
17
Years Until Maturity
19
21
23
25
14
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
15
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
16
Risk of Barbells and Ladders
 Interest
rate risk
 Reinvestment rate risk
 Reconciling interest rate and reinvestment
rate risks
17
Interest Rate Risk
 Duration
 The
increases as maturity increases
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
18
Interest Rate Risk (cont’d)
 Declining
interest rates favor a laddered
strategy
 Increasing
interest rates favor a barbell
strategy
19
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

20
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
21
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
22
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
23
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
24
Swaps
 Purpose
 Substitution
swap
 Intermarket or yield spread swap
 Bond-rating swap
 Rate anticipation swap
25
Purpose
 In
a bond swap, a portfolio manager
exchanges an existing bond or set of bonds
for a different issue
26
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
27
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
28
Substitution Swap (cont’d)
 Profitable
substitution swaps are
inconsistent with market efficiency
 Obvious
opportunities for substitution
swaps are rare
29
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
30
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
31
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
32
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 shortterm bonds
33
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
34
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
35
Bond Convexity
 The
importance of convexity
 Calculating convexity
 General rules of convexity
 Using convexity
36
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
37
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
38
The Importance
of Convexity (cont’d)
Bond Price
Greater Convexity
Yield to Maturity
39
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
40
Bond Price
The Importance
of Convexity (cont’d)
Error from using
duration only
Current bond
price
Yield to Maturity
41
Calculating Convexity
 The
percentage change in a bond’s price
associated with a change in the bond’s yield
to maturity:
dP  1 dP   1 d 2 P
Error
2 
 
dR     2 (dR )  
P  P dR   2 P dR
P

where P  bond price
R  yield to maturity
42
Calculating Convexity (cont’d)
 The
second term contains the bond
convexity:
1 d 2P
Convexity 
 2 (dR 2 )
2 P dR
43
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
44
Calculating Convexity (cont’d)
 Modified
duration is calculated as follows:
Macaulay duration
Modified duration 
1   Annual yield to maturity / 2  
45
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
46
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
47