23.2Duration

Duration is a concept that is closely related to DV01.The durationof a bond (or a

bond portfolio or cash flow stream), denoted DUR,is a weighted average of the wait-

ing times (measured in years) for receiving its promised future cash flows. The weight

on each time is proportional to the discounted value of the cash flow to be paid at

that time; that is, letting rdenote the yield (or discount rate) for the bond, Pdenote

the bond’s market price, and Cdenote the cash flow at date t,the duration DURof

t

the bond is

CCC
112 2. . .TT
(1r)r)2T
(1(1r)
DUR	(23.3)
CCC
12T
. . .
(1r)(1r)2T
(1r)

T

PV(C)

tt

P

t 1

Note that the weights, PV(C) P,always sum to 1.

t

We now explore some of the properties of duration.

The Duration of Zero-Coupon Bonds

The duration of a zero-coupon bond is the number of years to the maturity date of

the bond. Since a zero-coupon bond has only one cash flow, paid at maturity, the

weight on its maturity date is one. Hence, a 10-year zero-coupon bond has a dura-

tion of 10 years and a 6-year zero-coupon bond has a duration of 6 years. Aportfo-

lio of a group of 10-year zero-coupon bonds also would have a duration of 10 years.

All cash flows occur at year 10; hence, the 10-year timing of these cash flows

receives a weight of one.

The Duration of Coupon Bonds

It is useful to view a bond with coupons as a portfolio of zero-coupon bonds. For exam-

ple, a 10-year bond with a 5 percent coupon paid annually and a $100 face value can

be thought of as a 10-year zero-coupon bond with a $105 face value—the principal

plus the final coupon—plus nine other zero-coupon bonds, one for each of the first

9years, each with a face value of $5. Aflat term structure of interest rates implies that

the discount rate for each of the 10 cash flows is the same, and that duration is sim-

ply the (present value) weighted average of the durations of the cash flows that make

up the bond, as Example 23.6 shows.

Example 23.6:Computing the Duration of a Straight Coupon Bond

Compute the duration of a semiannual straight-coupon bond with a 2-year maturity.The bond

trades at par and has an 8 percent annualized coupon.Assume the term structure of inter-

est rates is flat.

Answer:Since the bond trades at par and the term structure of interest rates is flat, the

semiannually compounded discount rate is 8 percent.(See Result 2.2 in Chapter 2.)

•	The first coupon of $4.00 paid 6 months from now has a present value of $3.846 at8 percent and thus a weight equal to $3.846 bond price .03846.

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Chapter 23

Interest Rate Risk Management

827

•	The second coupon, 1 year from now, has a present value of $3.698 and a corresponding weight of .03698.

•	The third coupon, 1.5 years from now, has a present value of $3.556 and a corresponding weight of .03556.

•	The final cash flow, $104, has a present value of $88.900 and a weight of .889.

Hence, duration, the weighted average of the times the cash flows are paid, is computed as

.03846(.5 years) .03698(1 year) .03556(1.5 years) .88900(2 years) 1.89 years

Durations of Discount and Premium-Coupon Bonds

Exhibit 23.2 illustrates the duration of a coupon bond, a weighted average of the times at

which cash flows are paid, as the fulcrum (the black triangle) on the time scale where the

discounted valuesof the cash flows (principal plus coupon) balance.As Exhibit 23.3 in

comparison with Exhibit 23.2 indicates, the fulcrum in Exhibit 23.2 needs to be shifted to

the left to maintain a balance if the coupons increase because such an increase would also

increase their discounted values proportionately. Hence, other things being equal, premium

bonds have lower durations, and discount bonds, being more like zero-coupon bonds, have

higher durations.

How Duration Changes as Time Elapses

Exhibit 23.4 shows the duration of a straight coupon bond with semiannual payments as

time elapses, holding the yield to maturity constant. Note that as time elapses, the bond’s

duration increases at coupon dates. As the coupon is paid, the weights on all the other cash

flows are readjusted immediately. Between coupon dates, duration constantly decreases.

Durations of Bond Portfolios

The durations of bond portfolios are computed in the same manner as the durations

ofcoupon bonds. If we were to hold both 6-year zero-coupon bonds and 8-year

EXHIBIT23.2Duration as a Fulcrum

Cash flow present value

principal

coupon

1	2	3 4
		Year

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828	Part VIRisk Management EXHIBIT23.3Duration as a Fulcrum (with LargerCoupons)

Cash flow present value

principal

coupon

1	2	4
		Year

EXHIBIT23.4Duration as Time Elapses

Duration

3

2

1

				Years
.5	1 1.5	2 2.5 3	3.5 4

zero-coupon bonds, the duration of the bond portfolio would lie somewhere between

six and eight years. Suppose that the discounted value of the 6-year and 8-year

zero-coupon bonds is $10 million each. Then, since year 6 would have the same weight

as year 8 and the weights add to 1, each weight would be .5 and the duration of the

$20 million portfolio of the two bonds would be 7 years [ .5(6 years) .5(8 years)].

This result can be generalized to any portfolio of bonds as indicated below.

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Chapter 23

Interest Rate Risk Management

829

Result 23.3

Assuming that the term structure of interest rates is flat, the duration of a portfolio of bondsis the portfolio-weighted average of the durations of the respective bonds in the portfolio.

How Duration Changes as Interest Rates Increase

Result 23.3 implies that an increase in the yield to maturity of the bond decreases dura-

tion (compare Exhibit 23.5 to Exhibit 23.4). If we think of a coupon-paying bond as a

portfolio of zero-coupon bonds, then an increase in the bond’s yield to maturity reduces

the weight of the later cash flow payments proportionately more than it reduces the

weight of the early cash flows.

Result 23.3 also provides insights into how to alter the duration of a corporation’s

debt. Example 23.7 demonstrates how this is achieved in a hypothetical situation

involving Disney.

Example 23.7:Changing the Duration of Corporate Liabilities

Assume that Disney’s debt obligations have a market value of $2 billion and a duration of

five years.Half of this is a 10-year note with a duration of six years.The Disney treasurer

has decided that the company should target a debt duration of four years instead of five.

Assume that it is possible to issue 4-year par straight-coupon notes, which have a duration

of three years.If the proceeds from issuing these notes are used to retire 10-year notes,

there will be a reduction in the duration of Disney’s liabilities.Assuming a flat yield curve,

how many dollars of 4-year notes should Disney issue?

Answer:Disney’s existing liabilities consist of $1 billion in 10-year notes with a duration

of six years and, by Result 23.3, $1 billion in other liabilities with a duration of four years.

IfDisney issues $xin notes with three years’duration, and uses the proceeds to retire the

10-year notes, the duration of its new liability structure will be

$x$1 billion$^x$1 billion

DUR 3 years 6 years 4 years

$2 billion$2 billion$2 billion

2

DUR 4 years when x $3billion.Thus, issuing $666,666,667 in 3-year notes and using

the proceeds to retire two-thirds of the 10-year notes puts Disney at its target liability duration.

EXHIBIT23.5Duration as a Fulcrum (with a HigherDiscount Rate)

Cash flow present value

principal

coupon

1	2	3 4
		Year

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830	Part VIRisk Management

23.3	Linking Duration to DV01

This section develops formulas that link duration to DV01.First, it shows that the dura-

tion of a bond is related to the interest rate risk of the bond.

Duration as a Derivative

Duration is a useful tool for managing the interest rate risk of bond portfolios. This sug-

gests that duration is related to the derivative of a bond’s price with respect to interest

rates. Such a derivative also will enable us to relate duration to DV01.To illustrate the

relationship between duration and DV01,consider a 2-year zero-coupon bond with a face

value of $100 and a continuously compounded yield to maturity of r.The bond’s price

is P $100e2^rand its duration is two years. Using calculus, the percentage change in

the bond’s price dP Pfor a change in the continuously compounded yield-to-maturity is

dP PdP11
$100(2e2r) 2	(23.4)
dre2r
drP$100

Thus, 2, the negative of the bond’s duration, is the percentage sensitivity of the

bond’s price to changes in the bond’s continuously compounded yield.

This derivative property can be generalized. When the term structure of interest

rates is flat, duration always can be viewed as minus the percentage change in the value

of the bond (or portfolio) with respect to changes in its continuously compounded yield

to maturity. For example a bond maturing in Tyears, with cash flows of Cat date t,

t

t 1,..., T,has a price of

T
P Cert	(23.5)
t
t 1

if ris the bond’s continuously compounded yield to maturity.

The percentage change in Pwith respect to the continuously compounded yield is

T
dP P1
tCert DUR	(23.6)
drPt
t 1

This is the negative of duration because the weight assigned to each relevant date (the

ts) is the negative of the discounted value of the cash flow at that time divided by the

discounted value of the entire bond P.

To alter equation (23.6) for different compounding frequencies, note the following

formulas:

•	If annually compounded yields had been used in place of continuously compounded yields for this analysis, the derivative corresponding to equation (23.6) would be

T

dP P11CDUR

t

t

dr1rP(1r)^t1

r

t 1

indicating that duration is the negative of (1 r) times the percentage price

change for a small change in the yield.

•For semiannually compounded yields, the derivative corresponding to equation

(23.6) would be

2T

dP P11tCDUR

t

drr 2r 2)^t

1P2(11r2

t 1

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Chapter 23

Interest Rate Risk Management

831

indicating that duration is the negative of (1 r 2) times the percentage price

change for a small change in the yield.

•For monthly compounded yields, the derivative corresponding to equation

(23.6) would be

12T

dP P11tCDUR

t

drr 12r 12)^t

1P12(11r12

t 1

indicating that duration is the negative of (1 r 12) times the percentage

price change for a small change in the yield.

•As the compounding frequency mbecomes infinite, the denominator under

DUR, (1r m), converges to 1, which leads to the formula in equation

(23.6).

Formulas Relating Duration to DV01

The relation between duration and DV01is straightforward if the 01in DV01is defined

as a continuously compounded rate. In this case, DV01is the product of .0001 and

the derivative of the value of the bond with respect to a shift in the bond’s continu-

ously compounded yield; that is

dP

DV01 .0001

dr

Equation (23.6) implies that duration is the derivative of the percentage change in the value

of the bond with respect to the (continuously compounded) yield to maturity; that is

1dP

DUR

Pdr

Solving either of the last two equations for dP drand substituting its equivalent value

into the other equation gives an equation that relates DV01to duration

DV01 DURP.0001	(23.7a)
Equation (23.7a) suggests the following result:

Result 23.4

Because DV01can be translated into duration and vice versa, DV01and duration are equiv-alent as tools both for measuring interest rate risk and for hedging.

Modified Duration.If DV01is based on rates that are compounded mtimes a year

instead of continuously, then the formula relating DV01to duration is modified as follows

DUR
DV01 rP.0001	(23.7b)
1
m

The term in brackets is known as modified duration.

Effective Duration forAssets and Liabilities with Risky Cash Flow Streams.

Duration is more complicated to implement in a corporate setting without first linking

duration to DV01.Because most corporate cash flows are risky, weighting the maturi-

ties of uncertain cash flows to come up with a measure of the future timing of cash

flows makes little sense. However, duration, like DV01,is a measure of the interest

rate sensitivity of a cash flow, which can alternatively be obtained from a factor model.

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Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

832Part VIRisk Management

Thus, one can compute an effective durationof a corporate asset or liability with risk

(generated both by the uncertain cash flow stream and by changes in the discount rate(s)

for the cash flows in the stream) by first estimating its DV01as an interest rate sensi-

tivity in a factor model and then inverting equation (23.7b) to obtain DUR.This effec-

tive duration tells us that the corporate asset or liability is of the same sensitivity to

interest rate risk per dollar invested as a riskless zero-coupon bond of maturity DUR.

Hedging with DV01s or Durations

The last section provided formulas that link DV01directly to duration. Hence, both

duration and DV01are equally good tools for hedging bond portfolios. Recall that a

perfect hedge makes the new portfolio, including the hedge investment, have a DV01

of zero. To obtain a DV01of zero, the ratio of the duration of the unhedged bond port-

folio (DUR) to the duration of its hedge portfolio (DUR) must be inversely propor-

BH

tional to the ratio of the respective market values of the bonds;¹⁰that is

DURP

H^B

DURP

BH

If this property holds, the combination of the unhedged bond position and the short

position in the hedge has a duration (and DV01) of zero.

When duration is based on noncontinuously compounded yields, this relation is still

valid because the durations are both multiplied by the same constant. Of course, our

caveat that perfect hedging with DV01s occurs only when the term structure of inter-

est rates is flat applies here as well.

Example 23.8:Using Duration to Form a Riskless Hedge Position

Giuliana holds $1,000,000 (face value) of 10-year zero-coupon bonds with a yield to matu-

rity of 8 percent compounded semiannually.How can she perfectly hedge this position with

a short position in 5-year zero-coupon bonds with a yield to maturity of 8 percent (com-

pounded semiannually)?

Answer:The ratio of the durations of the two bonds is 10/5 2.The ratio of their mar-

ket values should therefore be 1/2.The market value of the 10-year bonds is

$1,000,000

$456,386.95

1.04²⁰

The market value of the 5-year bonds is

x

1.04¹⁰

For the market value of the 10-year bond position to be half the value of the 5-year posi-

tion, xmust solve

1x

$456,386.95

21.04¹⁰

or	x $1,351,128.34

¹⁰From

equation (23.7a), DV01is zero when

(DURP.0001) (DURP.0001) 0

BBHH

This equation, when rearranged, proves the result.

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Chapter 23

Interest Rate Risk Management

833

Hence selling short $1,351,128.34 (face value) of the 5-year bonds hedges the 10-year bond

position.

It is easy to compute what would have happened if the position in the 10-year

zero-coupon bonds was left unhedged in the last example. When rearranged, equation

(23.6) says

dP

DURdr

P

suggesting that a 10 basis point increase in interest rates (for example, from 9 percent

to 9.1 percent) would decrease the value of the bond portfolio by

.001DUR .00110 .01, or 1%

Hence, the $1 million (face amount) bond investment in the last example, which cost

$456,387, would decline by 1 percent of $456,387, or $4,564, if left unhedged. Under

this same interest rate change scenario, the $1.35 million (face value) of 5-year zero-

coupon bonds also would decline in value by $4,564. Hence, Giuliana constructs a port-

folio that has no interest rate sensitivity by selling short $1.35 million of the 5-year

bonds.

Duration targeting is often used to perfectly hedge pension fund liabilities. Apen-

sion that funds its liability for retirement obligations with default-free bonds would

want the bonds to have the same interest rate sensitivity as the liabilities. An over-

funded pension plan with this property guarantees that there will be no shortage of

funds for the pension liabilities as a result of interest rate changes. Example 23.9 indi-

cates how to target the duration of pension assets to hedge pension liabilities with this

purpose in mind.

Example 23.9:Changing the Duration of Pension Fund Assets

As the new manager of the pension fund of the University of California Retirement System,

assume that you have analyzed the defined benefits of the plan and computed that the fund’s

liabilities amount to an $8 billion market value obligation with a duration of 12 years.Unfor-

tunately, while the fund has $9 billion in assets, largely consisting of bonds, their duration is

only 8 years.This means that a steep decline in interest rates may increase the present

value of the obligations by $1 billion more than such a decline increases the value of the

fund’s assets.While keeping a $1 billion surplus in the pension fund, how can a self-financing

investment in 5-year Treasury notes, with a duration of 4 years, and 30-year Treasury bonds,

with a duration of 10 years, eliminate this problem?

Answer:If xdenotes the amount invested in 30-year bonds and an equivalent dollar

amount of 5-year bonds are sold short, the duration of the pension fund assets becomes

xx

DUR 8 years 10 years 4 years

$9 billion$9 billion

The ratio of the market values of the assets to the liabilities is 9/8.Hence, the duration of

2

the assets should be 103years, or 8/9 the 12-year duration of the liabilities to perfectly

²3$4 billion.Thus,

hedge interest risk.Solving for xabove with DUR 10years implies x

buying $4 billion in 30-year bonds and selling short $4 billion in 5-year bonds will result in

the interest rate sensitivity of the pension fund assets matching that of the liabilities.

Example 23.9 points out how to eliminate interest rate sensitivity by matching

assets and liabilities in a particular way. The next section examines how to carry this

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Strategy, Second Edition

834Part VIRisk Management

idea forward through time in order to fix an amount available for payment at some

horizon date.