Chapter 16

Managing Bond Portfolios

16-1

Outline• Interest rate risk• Duration• Convexity• Passive Bond Management

– Bond index fund– Immunisation

• Active Bond Management– Horizontal analysis

• Interest rate swaps• Financial engineering and interest rate derivatives

16-2

• Active strategy– Trade on interest rate predictions– Trade on market inefficiencies

• Passive strategy– Control risk– Balance risk and return

Basic Strategies

16-3

• Inverse relationship between price and yield• An increase in a bond’s yield to maturity results in a

smaller price decline than the gain associated with a decrease in yield

• Long-term bonds tend to be more price sensitive than short-term bonds

Bond Pricing Relationships

16-4

• As maturity increases, price sensitivity increases at a decreasing rate

• Price sensitivity is inversely related to a bond’s coupon rate

• Price sensitivity is inversely related to the yield to maturity

Bond Pricing Relationships (cont’d)

16-5

• A measure of the effective maturity of a bond• The weighted average of the times until each

payment is received, with the weights proportional to the P.V. of the payment

• Duration is shorter than maturity for all bonds except zero coupon bonds

• Duration is equal to maturity for zero coupon bonds

Duration

16-6

t tt

w CF y ice ( )1 Pr

twtDT

t

1

tperiodforFlowCashCFt

Duration: Calculation

16-7

8%Bond

Timeyears

Payment PV of CF(10%)

Weight C1 XC4

.5 40 38.095 .0395 .0197

1 40 36.281 .0376 .0376

1.5

2.0

40

1040

sum

34.553

855.611

964.540

.0358

.8871

1.000

.0537

1.7742

1.8852

Duration Calculation: Spreadsheet 16.1

16-8

Duration

• Is a simple statistic of the effective maturity of portfolio

• Is essential for immunisation• Is a measure of the interest rate sensitivity of the

portfolio

16-9

• Price change is proportional to duration and not to maturity

P/P = -D x [(1+y) / (1+y)]

D* = modified duration

D* = D / (1+y)

P/P = - D* x y

Duration/Price Relationship

16-10

Rules for Duration

Rule 1: The duration of a zero-coupon bond equals its time to maturity

Rule 2: Holding maturity constant, a bond’s duration is higher when the coupon rate is lower

Rule 3: Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity

Rule 4: Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower

16-11

Rules for Duration (cont’d)

Rules 5: The duration of a level perpetuity is equal to:

Rule 6: The duration of a level annuity is equal to:

1)1(

1

Ty

T

y

y

y

y)1(

16-12

Rules for Duration (cont’d)

Rule 7: The duration for a corporate bond is equal to:

yyc

ycTy

y

yT

]1)1[(

)()1(1

16-13

Yield

Price

Duration

Pricing Error from convexity

Duration and Convexity

16-14

Why Convexity?

• Duration approximation understate the value of the bond– Underestimate the increase in bond price and overestimate

the decrease in price

• The true price yield relationship has curvature called convexity

• Convexity improves the duration approximation

16-15

Correction for Convexity

n

tt

t tty

CF

yPConvexity

1

22

)()1()1(

1

Correction for Convexity:

])([21 2yConveixityyD

P

P

16-16

Duration and Convexity of Callable Bonds

• Negative convexity• Price yield curve exhibits an unattractive asymmetry.• Asymmetry due to the option to call back retained by

the issuer• Convexity prediction worse than the duration

approximation• Effective Duration: -(ΔP/P)/Δr

16-17

Passive Management

• Bond-Indexing– Index may have a large number of securities– Thinly traded bonds– Rebalancing problem

• Cellular Approach: subclasses of bonds– On the basis of maturity and issuer– On the basis of the coupon rate and the credit risk

16-18

Bond Index Funds

16-19

• Immunisation of interest rate risk:– Net worth immunisation

Duration of assets = Duration of liabilities

– Target date immunisationHolding Period matches Duration

– Rebalancing

• Cash flow matching and dedication

Passive Management

16-20

Duration Matching

• Balances the difference between:- the reinvestment risk and price risk- Rebalancing required when interest rate and asset duration

changes- Rebalancing is a continuous exercise even if interest rate is

unchanged

16-21

Why Active Management

• Duration matching immune portfolio only for parallel shift in the yield curve

• Immunisation is inappropriate goal in inflationary environment

• Need to characterise portfolio rebalancing activity through bond swaps

16-22

• Substitution swap• Inter-market swap• Rate anticipation swap• Pure yield pickup• Tax swap

Active Management: Swapping Strategies

16-23

Maturity

Yield to Maturity %

3 mon 6 mon 9 mon

1.5 1.25 .75

Yield Curve Ride

16-24

Contingent Immunisation

• A combination of active and passive management• The strategy involves active management with a floor

rate of return• As long as the rate earned exceeds the floor, the

portfolio is actively managed• Once the floor rate or trigger rate is reached, the

portfolio is immunised

16-25

Interest Rate Swaps

• Contract between two parties to exchange a series of cash flows

• One party pays a fixed rate and receives a variable rate

• One party pays a variable rate and receives a fixed rate

• Payments based on notional principal

16-26

Swap Example Figure 16-11

Swap Dealer Company BCompany A

LIBOR LIBOR

LIBOR

7%

6.95% 7.05%

16-27

Financial Engineering

• Inverse Floater– Performs poorly when interest rate rise and vice versa

• Created synthetically by allocating the cash flows from a fixed-rate security into two derivative securities

16-28

Summary

• Default free bonds also have interest rate risk• Life of the bond and sensitivity to interest rate

changes are related• Duration measures the average life of a bond• Convexity is the curvature of the bond’s price-yield

relationship

16-29

Summary

• Investors can immunise their portfolio against the interest rate changes

• Net-worth of a fixed-income portfolio can be immunised

• Duration of Assets = Duration of Liabilities• Periodical rebalancing or cash flow matching• Interest rate swap

16-30