Bond™s Duration Is Handy Guide on Rates

Suppose you buy a 10-year Treasury note today at a time in that note™s life, potential price changes as rates

yield to maturity of 6% and interest rates shoot up to go down or up would be about equally offset by conse-

8%. What happens to your investment? quent changes in the amount of interest-on-interest

that would be received.

A. You lose money.

As a result, the bond™s total return will about equal

B. You make money. its initial yield to maturity, even if rates change.

Duration is a reassuring feature for investors who

C. Nothing happens.

have expenses coming due in the future”for retire-

D. All of the above.

ment or tuition payments, for instance”that they need

The answer: D. All of the above. to cover with the proceeds of their bond investments.

How is that possible? The trick is how long you hold By making sure the duration of their investments

the investment. roughly matches the due date of their expenses, they

In the short run, you lose money. Since interest rates can avoid being caught off guard by adverse rises in in-

and bond prices move inversely to one another, higher terest rates.

rates mean the value of your bond investment withers

when rates go up. For a 10-year Treasury yielding 6%, GAUGE OF RISK

a two percentage-point rise in rates would cause the

But the best thing about duration may be that it pro-

value of your principal to sink by roughly 14%.

vides an extremely handy gauge of interest-rate risk in

However, if you hold the note, rather than selling it,

a given bond or bond fund. To figure out how much

you™ll get to reinvest the interest received from it at the

prices will move in response to rate changes, simply

new, higher 8% rate. Over time, this higher “interest on

multiply the percentage change in rates by the duration

interest” adds up, allowing you not only to offset your

of the bond or bond fund and, voila, you have a pretty

initial loss of principal but also to profit more than if

good estimate of what to expect.

rates had never moved at all.

For instance, if rates go from 6% to 8%, a 10-year

Perhaps the best way to judge a bond™s interest-rate

Treasury note with a duration of 7.4 will take a price hit

sensitivity is to get a handle on its “duration.” Duration

of about 13.5%, or a bit less than two percentage

is one measure of a bond™s life. It™s that sweet spot,

points times the duration.

somewhere between the short term and the long term,

where a bond™s return remains practically unchanged,

no matter what happens to interest rates.

Source: Barbara Donnelly Granito, “Bond™s Duration Is Handy Guide

BOND™S DURATION on Rates,” The Wall Street Journal, April 19, 1993. Reprinted by

permission of Dow Jones & Company, Inc. via Copyright Clearance

The duration of a 10-year Treasury note yielding 6% in Center, Inc. © 1993 Dow Jones & Company, Inc. All Rights Reserved

today™s market is between seven and 71„2 years. By that Worldwide.

obligation were not duration-matched across the interest rate shift, so the position was not

fully immunized.

This example demonstrates the need for rebalancing immunized portfolios. As interest rebalancing

rates and asset durations continually change, managers must rebalance, that is, change the Realigning the

composition of, the portfolio of fixed-income assets to realign its duration with the duration of proportions

the obligation. Moreover, even if interest rates do not change, asset durations will change of assets in a

portfolio as needed.

solely because of the passage of time. Recall from Figure 10.3 that duration generally de-

creases less rapidly than maturity as time passes, so even if an obligation is immunized at the

outset, the durations of the asset and liability will fall at different rates. Without portfolio re-

balancing, durations will become unmatched and the goals of immunization will not be real-

ized. Therefore, immunization is a passive strategy only in the sense that it does not involve

attempts to identify undervalued securities. Immunization managers still actively update and

monitor their positions.

355

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

356 Part THREE Debt Securities

An insurance company must make a payment of $19,487 in seven years. The market interest

rate is 10%, so the present value of the obligation is $10,000. The company™s portfolio man-

10.2 EXAMPLE ager wishes to fund the obligation using three-year zero-coupon bonds and perpetuities paying

annual coupons. (We focus on zeros and perpetuities to keep the algebra simple.) How can the

Constructing an

manager immunize the obligation?

Immunized

Immunization requires that the duration of the portfolio of assets equal the duration of the

Portfolio

liability. We can proceed in four steps:

Step 1. Calculate the duration of the liability. In this case, the liability duration is simple to

compute. It is a single-payment obligation with duration of seven years.

Step 2. Calculate the duration of the asset portfolio. The portfolio duration is the weighted

average of duration of each component asset, with weights proportional to the funds

placed in each asset. The duration of the zero-coupon bond is simply its maturity,

three years. The duration of the perpetuity is 1.10/.10 11 years. Therefore, if the

fraction of the portfolio invested in the zero is called w, and the fraction invested in

the perpetuity is (1 w), the portfolio duration will be

w w)

Asset duration 3 years (1 11 years

Step 3. Find the asset mix that sets the duration of assets equal to the seven-year duration of

liabilities. This requires us to solve for w in the following equation

w w)

3 years (1 11 years 7 years

This implies that w 1/2. The manager should invest half the portfolio in the zero

and half in the perpetuity. This will result in an asset duration of seven years.

Step 4. Fully fund the obligation. Since the obligation has a present value of $10,000, and

the fund will be invested equally in the zero and the perpetuity, the manager must

purchase $5,000 of the zero-coupon bond and $5,000 of the perpetuity. (Note that

the face value of the zero will be $5,000 (1.10)3 $6,655.)

Even if a position is immunized, however, the portfolio manager still cannot rest. This is

because of the need for rebalancing in response to changes in interest rates. Moreover, even if

rates do not change, the passage of time also will affect duration and require rebalancing. Let

us continue Example 10.2 and see how the portfolio manager can maintain an immunized

position.

Suppose that one year has passed, and the interest rate remains at 10%. The portfolio man-

ager of Example 10.2 needs to reexamine her position. Is the position still fully funded? Is it still

10.3 EXAMPLE immunized? If not, what actions are required?

First, examine funding. The present value of the obligation will have grown to $11,000, as

Rebalancing

it is one year closer to maturity. The manager™s funds also have grown to $11,000: The zero-

coupon bonds have increased in value from $5,000 to $5,500 with the passage of time, while

the perpetuity has paid its annual $500 coupons and remains worth $5,000. Therefore, the ob-

ligation is still fully funded.

The portfolio weights must be changed, however. The zero-coupon bond now will have a du-

ration of two years, while the perpetuity duration remains at 11 years. The obligation is now

due in six years. The weights must now satisfy the equation

w w)

2 (1 11 6

which implies that w 5/9. To rebalance the portfolio and maintain the duration match, the

manager now must invest a total of $11,000 5/9 $6,111.11 in the zero-coupon bond.

This requires that the entire $500 coupon payment be invested in the zero, with an additional

$111.11 of the perpetuity sold and invested in the zero-coupon bond.

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

EXCE L Applications www.mhhe.com/bkm

> Immunization

The Excel immunization model allows you to analyze any number of time-period or holding-period

immunization examples. The model is built using formulas for bond duration, which allow the

investigation of any maturity bond without building a table of cash flows. (This model contains

sample relationships similar to those displayed in Table 10.4.)

You can learn more about this spreadsheet model by using the interactive version available on

our website at www.mhhe.com/bkm.

A B C D E F G H

1 Holding Peiod Immunization

2

3 YTM 0.0800 Mar Price 1000.00

4 Coupon R 0.0800

5 Maturity 6

6 Par Value 1000.00

7 Holding P 5

8 Duration 4.9927

9

10

11 If Rates Increase by 200 basis points If Rates Increase by 100 basis points

12 Rate 0.1000 Rate 0.0900

13 FV of CPS 488.41 FV of CPS 478.78

14 SalesP 981.82 SalesP 990.83

15 Total 1470.23 Total 1469.60

16 IRR 0.0801 IRR 0.0800

17

18

19

20 If Rates Decrease by 200 basis points If Rates Decrease by 100 basis points

21 Rate 0.0600 Rate 0.0700

22 FV of CPS 450.97 FV of CPS 460.06

23 SalesP 1018.87 SalesP 1009.35

24 Total 1469.84 Total 1469.40

25 IRR 0.0801 IRR 0.0800

Of course, rebalancing of the portfolio entails transaction costs as assets are bought or sold,

so continuous rebalancing is not feasible. In practice, managers strike some compromise be-

tween the desire for perfect immunization, which requires continual rebalancing, and the need

to control trading costs, which dictates less frequent rebalancing.

<

4. What would be the immunizing weights in the second year if the interest rate were Concept

to fall to 8%?

CHECK

Cash Flow Matching and Dedication

The problems associated with immunization seem to have a simple solution. Why not simply

buy a zero-coupon bond that provides a payment in an amount exactly sufficient to cover the

357

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

358 Part THREE Debt Securities

projected cash outlay? This is cash flow matching, which automatically immunizes a port-

cash flow

folio from interest rate movements because the cash flow from the bond and the obligation

matching

exactly offset each other.

Matching cash flows

Cash flow matching on a multiperiod basis is referred to as a dedication strategy. In this

from a fixed-income

case, the manager selects either zero-coupon or coupon bonds that provide total cash flows

portfolio with those

of an obligation. that match a series of obligations in each period. The advantage of dedication is that it is a

once-and-for-all approach to eliminating interest rate risk. Once the cash flows are matched,

dedication there is no need for rebalancing. The dedicated portfolio provides the cash necessary to pay

strategy the firm™s liabilities regardless of the eventual path of interest rates.

Cash flow matching is not widely pursued, however, probably because of the constraints it

Refers to multiperiod

imposes on bond selection. Immunization/dedication strategies are appealing to firms that do

cash flow matching.

not wish to bet on general movements in interest rates, yet these firms may want to immunize

using bonds they believe are undervalued. Cash flow matching places enough constraints on

bond selection that it can make it impossible to pursue a dedication strategy using only

“underpriced” bonds. Firms looking for underpriced bonds exchange exact and easy dedica-

tion for the possibility of achieving superior returns from their bond portfolios.

Sometimes, cash flow matching is not even possible. To cash flow match for a pension

fund that is obligated to pay out a perpetual flow of income to current and future retirees, the

pension fund would need to purchase fixed-income securities with maturities ranging up to

hundreds of years. Such securities do not exist, making exact dedication infeasible. Immu-

nization is easy, however. If the interest rate is 8%, for example, the duration of the pension

fund obligation is 1.08/.08 13.5 years (see Rule 5 above). Therefore, the fund can immunize

its obligation by purchasing zero-coupon bonds with maturity of 13.5 years and a market

value equal to that of the pension liabilities.

>

5. a. Suppose that this pension fund is obligated to pay out $800,000 per year in

Concept

perpetuity. What should be the maturity and face value of the zero-coupon

CHECK bond it purchases to immunize its obligation?

b. Now suppose the interest rate immediately increases to 8.1%. How should the

fund rebalance in order to remain immunized against further interest rate

shocks? Ignore transaction costs.

6. How would an increase in trading costs affect the attractiveness of dedication ver-

sus immunization?

10.3 CONVEXITY

Duration clearly is a key tool in bond portfolio management. Yet, the duration rule for the im-

pact of interest rates on bond prices is only an approximation. Equation 10.3, which we repeat

here, states that the percentage change in the value of a bond approximately equals the prod-

uct of modified duration times the change in the bond™s yield:

P

D* y (10.3)

P

This rule asserts that the percentage price change is directly proportional to the change in the