Managing Bond Portfolio

Objectives:

• Analyze the features of a bond that affect the sensitivity of its price to interest rates.

• Compute the duration of bonds.

• Formulate fixed-income immunization strategies for various investment horizons.

• Analyze the choices to be made in an actively managed fixed-income portfolio.

16.1 INTEREST RATE RISK

Interest rate risk is uncertainty related to bond prices due to uncertainty in interest rate

Interest rate sensitivity:

•Time to maturity

•Coupon rate

Inverse relationship between price and yield Long-term bonds tend to be more price

sensitive than short-term bonds Interest rate risk is inversely related to

bond’s coupon rate

Change in price of 8% coupon bond < change in price of 0% coupon bond or 8% bond has higher interest rate risk than 0% bond

We also know that long-term bond has higher interest rate risk than short-term bond.

This implies that 0% bond is a longer term investment than 8% bond.

This implies that regular maturity is not good enough to tell about the short term/long term nature (or interest rate sensitivity) of the bonds.

We need a maturity that can tell 0% bond is really longer term investment than 8% bond

Hence we have concept of effective maturity or duration that takes into account both regular maturity and coupon payments.

8% coupon bondTime 0 1 2 3 ..... 20cash flow 40 40 40 40+1000 0% coupon bondTime 0 1 2 3 ..... 20cash flow 0 0 0 0+1000

If we view a bond is a portfolio of all coupon payments and principal payment, and each payment has its own maturity, and the effective maturity of the bond is the weighted average of all maturity of all payments, then obviously, the maturity of 0% coupon bond is different from maturity of 8% coupon bond.

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 present value of the payment

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

Duration is equal to maturity for zero coupon bonds

Measures the effective maturity by weighting the payments by their proportion of the bond value.

where t =1, 2, 3, ... T are the times to maturity of payments

Price Bond

1/CF

Price Bond

CFPV ttt

t

yw

y is the bond's yield to maturity (current market rate)

D t wtt

T

1

A pension plan is obligated to make disbursements of $1 million, $2 million, and $1 million at the end of each of the next three years, respectively. Find the duration of the plan’s obligations if the interest rate is 10% annually.

Duration measure does three things:

• It measures the effective average maturity of a bond.

• It measures interest rate sensitivity correctly.

• It provides the necessary information for immunization.

Sensitivity of prices to interest rate changes:

y

yD

P

P

1

where y is the yield to maturity

yDP

P

* duration modified is

1 and

*

*

D

y

DD

Example: The duration for a bond (6% coupon rate, semi-annual payment, 2 years to maturity), currently priced at $929.08, with a yield-to-maturity (YTM) of 10% is 1.91061 years. If interest rates rise by 0.5 percentage points (50 basis points), what will be the percentage change in the price of the bond?

• If interest rates rise by 0.1 percentage points (10 basis points), what will be the percentage change in the price of the bond?

The relationship between bond prices and yields is not linear

Duration rule is a good approximation for only small changes in bond yields

n

tt

t tty

CF

yPConvexity

1

22

)()1()1(

1

Correction for Convexity:

21 [ ( ) ]2P

D y Convexity yP

Determinants of a bond’s price sensitivity to interest rate changes:

•the time to maturity (Duration not always increasing in time to Maturity)

•the coupon rate(Duration always decrease with high Coupon)

•the yield to maturity(Duration always decrease if YTM increase)

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

Rules 5 The duration of a level perpetuity is equal to: (1+y) / y

Key concept in bond management Simple summary measure of effective maturity of bond. It

is the effective maturity that tells the short-term/long-term nature of bond, not regular maturity

Measure of interest sensitivity of a bond portfolio◦ Bond A: 8% semi-annual coupon, 2 years to maturity, D = 1.8852

Bond B: 0% coupon, 2 years to maturity, D = 2Bond B has more interest rate risk than bond A

◦ Bond A: 8% semi-annual coupon, 2 years to maturity, D = 1.8852Bond B: 0% coupon, 1.8852 years to maturity, D = 1.8852Bond B and Bond A has same interest rate risk

• It provides the necessary information for immunization.

Longest-duration security provides the maximum price variation

If you expect a decline in interest rates, increase the average duration of your bond portfolio to experience maximum price increase

If you expect an increase in interest rates, reduce the average duration to minimize your price decline

FIN 8330 Lecture 4 9/6/2007 25

10.2 PASSIVE BOND MANAGEMENT

1. Net Worth Immunization (Present) (e.g. Banks: Asset/Liability Management)

2. Target Date Immunization (Future) (e.g. Pension Funds: meet future obligations)

Takes prices as given and tries to control the risk of the fixed-income portfolio.

Measures:

Net Worth Immunization: Match duration of asset and liabilities by adjusting

their maturity structure (Gap Management)

Target Date Immunization:Set the duration of a portfolio equal to the target date.

This guarantees that at this date reinvestment risk and price risk exactly cancel out.

Example. An insurance company issue a 5-years Guaranteed investment contract (GIC) at 8%, nominal value $10,000. The insurance company decides to meet this obligation by investing $10,000 in 8% annual coupon bonds with maturity in 6yrs. Can the firm meets its obligation at time 5?What if interest rate drops to 7% ?What if interest rate increases to 9% ?

An insurance company must make payments to a customer of $10 mil in 1 year and $4 mil in 5 years. The market interest rate is 10%

a. If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?

b. What must be the face value and market value of that zero-coupon bond?

Given a liability currently worth L and with duration DL Match it with an asset currently worth L and with

duration DL. This guarantees that, for small changes in the interest

rate the net worth will always be approximately zero.

Current Value of Asset and Liabilities

Interest rate

Current of Coupon Bond (Asset)(YTM=8%)

Present Value of CIG (Liability)(YTM=8%)

8%=YTM

Even if a position is immunized, there is still need for rebalancing for duration because of◦ Change in interest rate in market◦ Passage of time

Cannot rebalance continuously because of transaction cost involved

In practice, must establish a compromise between the desire for perfect immunization which requires continuous rebalancing and the need to control for trading cost which dictates less frequent rebalancing

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10.3 ACTIVE BOND MANAGEMENT

Sources of potential profits:

•Interest rate forecasts

•Identification of mispriced bonds

Substitution swap Intermarket swap Rate anticipation swap Pure yield pickup Tax swap

Substitution swap◦ an exchange of bond for nearly identical substitute

(coupon, maturity, quality, call features, etc)◦ Toyota bond, 20 years to maturity, 8% coupon, YTM =

8.05%◦ Honda bond, 20 years to maturity, 8% coupon, YTM =

8.15%◦ Honda looks more attractive

Intermarket swap◦ the yield spread between 2 sectors of bond market is

temporarily out of line. ◦ Example: currently, the yield spread between 10-year

T-bond and 10 year BBB corporate bond is 3%, the historical spread is 2%, should consider selling T-bond and buy BBB corporate bond

Rate anticipation swap◦ expected rate to fall, swap into bonds of longer

duration. Expected rate to rise, swap into shorter duration

Pure yield pickup◦ the strategy depends on the shape of the yield

curve in the market◦ If yield curve is upward sloping, should move into

long-term bonds to get higher yield. However, at the same time, your portfolio is exposed to higher interest rate risk.

Tax swap: ◦ Swap a capital gain bond to a capital loss bond to

avoid tax

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 immunized