23Risk Management

Learning Objectives

After reading this chapter, you should be able to:

1.Describe the concept of DV01,the dollar value of a one-basis-point decrease (in

original and term structure variations) and the concept of duration (in MacAuley,

modified, and present value variations).

2.Understand the relation between duration and DV01,and the formulas needed to

use either concept for hedging.

3.Implement immunization and contingent immunization strategies, using both

duration and DV01.

4.Explain the link between immunization and hedging, and how to use this link to

manage the asset base and capital structure of financial institutions.

5.Understand and compute convexity, and know how to use it properly.

In 1995, Orange County, California, declared bankruptcy. This bankruptcy can be

attributed to the investments of the county treasurer, Robert Citron. In late 1994,

between one-third and one-half of Citron’s investments were in “inverse floaters,”

floating rate bonds with cash flows that decline as interest rates increase and vice

versa. The inverse floaters paid 17 percent less twice the short-term interest rate, as

long as this number was positive. The value of these esoteric notes was about three

times more sensitive to interest rate movements than fixed-rate debt of comparable

maturity. Moreover, Orange County leveraged these positions through repurchase

agreements, making the positions three times larger in size than the cash spent on

them and therefore about nine times more sensitive to interest rate movements than

an unleveraged position in fixed-rate debt of comparable maturity. It did not take a

very large increase in interest rates to wipe out Orange County’s pool of investable

funds.¹

1

We are grateful to Richard Roll for providing us with some of the details of this bankruptcy.

818

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Strategy, Second Edition

Chapter 23

Interest Rate Risk Management

819

Chapter 22 described how to hedge many of the sources of risk that corporations

face. Here we complete the work in that chapter by analyzing how to change the

firm’s exposure to an interest rate risk factor. Interest rate risk deserves special treat-

ment because interest rates, in addition to their possible effect on future cash flows,

generally affect the present values of cash flows owing to their role in discounting. It

is this latter role that makes interest rate risk unique.

In this chapter, you will learn how interest rate risk is managed. The chapter

abstracts from the influence of interest rates on future cash flows by assuming that

future cash flows are certain. Hence, the risk analysis here focuses only on the role of

interest rates in determining present values.

The key to this analysis is understanding the relationship between the price of a

financial instrument with riskless future cash flows (that is, a default-free bond) and its

yield to maturity.²

We introduce two important concepts that describe this relationship,

DV01and duration,compare them to one another, and describe several important appli-

cations of each.

In addition to studying the use of these tools for measuring interest rate risk and

designing hedges, the chapter examines immunization, which is the management of a

portfolio of fixed income investments so that they will have a riskless value at some

future date, and the concept of convexity, which is a measure of the curvature in the

price-yield relationship. Convexity is frequently used and misused in bond portfolio

analysis and management. We will try to understand the proper use of convexity as

well as the pitfalls that lead to its misuse.

Obviously, bond portfolio managers and high-level financial managers in banks and

other financial institutions should be familiar with the tools introduced in this chapter.

However, it also is important for other individuals, such as corporate treasurers, to

acquaint themselves with these tools. For example, the debt issued by a corporation is

priced by the bond market. To properly analyze the firm’s debt financing costs, it is

essential for corporate treasurers to understand how interest rates affect bond prices. In

addition, corporate treasurers often have to fund, manage, or supervise the management

of a corporate pension fund. Frequently, the assets in such funds are debt instruments.

It would be difficult to meet the obligations of these pension funds without knowledge

of the interest rate risk of both the obligations and the debt instruments held by the

fund, as well as how to manage this risk. More generally, corporations typically target

a maturity structure of their debt, measured as the debt’s duration, which is tied to the

interest rate sensitivity of the debt.³

How to achieve that target is one subject of this

chapter.

In much of this chapter, we assume that there is a flat term structure of interest

rates. Hence, there is only one discount rate for risk-free cash flows of any maturity.

Later in the chapter, we relax this simplifying assumption and discuss how to extend

the analysis to deal with discount rates that vary with the maturity of the cash flow.

The results obtained for hedge ratios and immunization strategies in this chapter work

well in most realistic settings, despite being based on assumptions that, while virtu-

ous for the simplicity they add, are not entirely realistic and may even be logically

inconsistent.

²For expositional clarity, we will often use the generic term bondto refer to all debt instruments.

While we recognize that there are important legal and economic distinctions between bonds, notes, bills,

and bank debt, such distinctions are not critical for the analysis in this chapter.

³Chapters 16 and 21 broadly touched on the reasons why corporations might want short-term instead

of long-term debt.

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Strategy, Second Edition

820	Part VIRisk Management

23.1	The DollarValue of a One Basis Point Decrease (DV01)

Once corporate managers have measured the interest rate risk exposure of a corpora-

tion or project, they can reduce the risk exposure of equity holders by acquiring or issu-

ing bonds, the values of which move in an opposite direction from the value of the

corporation or project as interest rates change. To implement a strategy like this, the

financial manager needs to acquire familiarity with the tools of interest rate risk man-

agement.

One of the most important tools for understanding the relation between interest rates,

as measured by a bond’s yield-to-maturity, and bond prices is the “dollar value of a one

basis point (bp) decrease,” or DV01,⁴

which is a measure of how much a bond’s price

will increase in response to a one basis point decline in a bond’s yield to maturity. If

the yield and interest rates are the same—as they would be if there is a flat term struc-

ture of interest rates—then DV01s will be useful for estimating interest rate risk. In this

case, because high DV01bonds are more sensitive to interest rate movements than low

DV01bonds, they also are more volatile than low DV01bonds. In a corporate setting,

firms with values that have high DV01s are more exposed to interest rate risk. Hence,

if a corporation measures the risk exposure of its stock as having a negative DV01(for

example, share prices go up when interest rates rise), acquiring a bond position with a

positive DV01will reduce the interest rate exposure of the corporation’s equity.

In reality, yields and interest rates differ because the term structure is not flat, mak-

ing DV01an inexact hedging tool. DV01,however, works remarkably well as a hedg-

ing tool despite this potential problem, as the introduction to this chapter noted.

Methods Used to Compute DV01for Traded Bonds

The various ways to compute the DV01of a bond are outlined below:

Method 1: Price at a Yield One Basis Point Below Existing Yield Less Price at

Existing Yield.The method we will generally use to compute a bond’s DV01is the

negative of the change in its full price⁵

for a one basis point decreasein its

yield-to-maturity. One basis point is 1 100th of 1 percent, implying, for example, that

a one basis point decrease in a 10 percent yield to maturity is a decrease from 10.00

percent to 9.99 percent. This is a very small decrease because DV01is intended to

approximate, in absolute magnitude, the derivative of the bond price movement with

respect to the bond’s yield to maturity.This derivative is the slope of the price-yield

curve pictured in Exhibit 23.1.⁶DV01is determined by a one basis point decreasein

interest rates, so that the DV01is positive for most bonds.⁷

Method 2: Use Calculus to Compute the Derivative of the Price-Yield Relation-

ship Directly.Some bond market participants use calculus to define the DV01as the

negative of the slope of the price-yield curve. In this case, the DV01of a bond with a

price of Pis computed as

dP
DV01 .0001	(23.1)
dr

4

DV01is often referred to as the price value of a basis point (PVBP).

5

See Chapter 2 for a description of full and flat bond prices.

6

The derivative is negative because of the inverse relationship between price and yield.

⁷This avoids the awkwardness of always having to remember to place a minus sign in front.

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Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

Chapter 23Interest Rate Risk Management	821
EXHIBIT23.1Inverse Relation between Price and Yield

Price

Yield

In equation (23.1), r,the bond yield, is given in decimal form. This gives a

slightly different answer than method 1 because a one basis point decline, although

small, is not infinitesimally small, as would be the case with the derivative

calculation.

11

Method 3: Price at a Yield Basis Point Below Less Price at Basis Point Above

22

Existing Yield.To approximate method 2’s derivative calculation more accurately

than method 1, some practitioners compute DV01by subtracting the bond’s present

11

value at basis point above the current yield to maturity from its present value at

22

basis point below the current yield. As with method 1, this definition of DV01requires

only a calculator for its computation, thus avoiding the use of calculus.

Using a Financial Calculatorto Compute DV01 with Method 1.The differences

between these versions of DV01are negligible. We employ method 1 for most of the

examples in this chapter, as in Example 23.1.

Example 23.1:Computing DV01

Compute the DV01per $100 face value of a 20-year straight-coupon bond trading at par

(see Chapter 2 for a definition) with a semiannual coupon and an 8 percent yield to maturity.

Answer:Using a financial calculator or computing the present value by hand, we find

that at a discount rate of 7.99 percent, the 8 percent coupon bond should trade for approx-

imately $100.10.At a discount rate of 8 percent, it trades at $100.Thus, the DV01or dif-

ference is $0.10 per $100 of face value.

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Strategy, Second Edition

822Part VIRisk Management

Using DV01to Estimate Price Changes

One can use DV01to estimate the price change of a bond or a bond portfolio for small

changes in the level of interest rates, as measured by the bond’s yield. To obtain the

formula for this estimate, rearrange equation (23.1) as follows.

dP DV0110,000dr

Because 10,000 ris the yield change in number of basis points, the noncalculus equiv-

alent of this equation is

	P DV01(bp)	(23.2)
where

P the change in the bond’s price

bp the interest rate change (in basis points).

Example 23.2 demonstrates how to use this formula to estimate bond price changes.

Example 23.2:Estimating Bond Price Changes with DV01

A bond position has a DV01of $300.Its yield is currently 8.0 percent.Estimate the change

in the value of the bond position if the yield drops to 7.9 percent.

Answer:Using equation (23.2), a drop of 10 basis points gives a price increase of

P $300 (10) $3,000

Note from equation (23.2) that if the DV01is zero, the change in the price for a

small change in yield is zero. We will discuss how to use this insight for interest rate

hedging shortly.

DV01s of Various Bond Types and Portfolios

Bonds with long maturities tend to have higher DV01s than short-maturity bonds with

similar values. Zero-coupon bonds with long maturities tend to have low DV01s

because their prices are low. As a percentage of the price, however, the volatilities of

these bonds tend to be high.

DV01s are proportional to the size of a bond position. The DV01of a $1,000,000

position in a bond (face value or market value) is 10 times the DV01of a $100,000

position in that same bond. To obtain the DV01of a bond portfolio, add up the DV01s

of all the components in that portfolio.

Practitioners often scale DV01to be DV01per $100 of face value or per $1 mil-

lion of face value. This scaling makes it easy to compare individual bonds, but it is a

minor inconvenience when determining the sensitivity of a bond portfolio’s total value

to yield changes. To overcome this inconvenience, first convert the DV01s per $100 or

DV01s per $1 million into the DV01s of the total face values of each bond in the port-

folio. Then, add the DV01s of the components of the portfolio to obtain the DV01of

the portfolio itself. Keep in mind two helpful rules when computing the DV01of a

portfolio of bonds:

•	To convert a DV01per $100 face value to a DV01per actual face value,multiply the actual face value of the portfolio’s position in the bond by theDV01per $100 face value and then divide by $100.

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Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

Chapter 23

Interest Rate Risk Management

823

•	To convert a DV01per $1 million face value to a DV01per actual face value,multiply the actual face value of the portfolio’s position in the bond by theDV01per $1 million face value and then divide by $1 million.

Example 23.3 uses the first of these two rules to compute the DV01of a portfolio of

bonds.

Example 23.3:The DV01of a Portfolio of Bonds

Assume that the DV01of a 30-year U.S.Treasury bond is $0.10 per $100 face amount, the

DV01of a 10-year U.S.Treasury note is $0.06 per $100 face amount, and the DV01of a

5-year U.S.Treasury note is $0.04 per $100 face amount.Compute the DV01of a portfolio

that has $5 million (face value) of 30-year bonds, $8 million (face value) of 5-year notes, and

(a short position) –$16 million (face value) of 10-year notes.

Answer:Sum the products of (a) the face amounts over $100 and (b) the corresponding

DV01s per $100.The result is

$5 million$8 million$16 million

($.10) ($.04) ($.06)

DV01

$100$100$100

$5,000$3,200$9,600 $1,400

The negative DV01means that the position’s value increases when interest rates rise.

Using DV01s to Hedge Interest Rate Risk

DV01s are commonly used for interest rate hedging. This subsection discusses the hedg-

ing of interest rate risk in both investment management and corporate settings.

Hedging the Interest Rate Risk of a Bond Portfolio.Aperfectly hedged bond port-

folio is a portfolio with no sensitivity to interest rate movements. Result 23.1 charac-

terizes such a portfolio in terms of DV01.

Result 23.1

If the term structure of interest rates is flat, a bond portfolio with a DV01of zero has nosensitivity to interest rate movements.

Example 23.4 illustrates how to construct a portfolio with a DV01of zero.

Example 23.4:Using DV01s to Form Perfect Hedge Portfolios

Assume that changes in the yields of various maturity bonds are identical.How much of a

7-year bond, with a computed DV01of $0.05 per $100 of face value, should be purchased

to perfectly hedge the interest rate risk of the portfolio in Example 23.3, which has a DV01

of $1,400?

Answer:The portfolio to be hedged has a DV01of $1,400.The DV01of the portfolio

if we buy $xface amount of the 7-year bond is

$0.05x

$1,400

100

This equals zero when xis $2.8 million.Thus, a $2.8 million face amount of 7-year bonds

hedges the portfolio against interest rate risk.

Equation (23.2) suggests that for the unhedged portfolio constructed in Exam-

ple 23.3, a 10 basis point decrease in interest rates (for example, rates decrease from

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Strategy, Second Edition

824Part VIRisk Management

9.0 percent to 8.9 percent) would (approximately) result in a decline of $14,000 in

the portfolio’s value. However, after implementing the hedge in Example 23.4, the

DV01of the overall portfolio is zero, so the change in the hedged portfolio’s value

for a 10 bp decrease in rates is zero (approximately). In this case, the $14,000

decline in the formerly unhedged portfolio is offset by a $14,000 increase in the

$2.8 million of 7-year bonds.

Dynamically Updating the Hedge.The hedge in Example 23.4 needs to be con-

stantly readjusted because as time elapses or bond prices change, the DV01s of bonds

change. Thus, it will be necessary to sell or buy additional 7-year bonds as time

elapses to maintain a DV01of zero. In practice, to save on transaction costs, this

updating usually involves waiting until a critical threshold of positive or negative

DV01for the “hedged” portfolio is reached before rehedging. The size of the thresh-

old depends on the size of the transaction costs for rehedging and the investor’s aver-

sion to interest rate risk.

Managing Corporate Interest Rate Risk Exposure.Hedging corporate exposure to

interest rate risk is virtually identical to hedging the interest rate risk in a bond port-

folio. If we estimate a corporation’s interest rate risk exposure to be a DV01of

$1,400, then $2.8 million of the 7-year bonds would perfectly hedge the corporation

against interest rate risk.

It is also possible to use the mathematics of Example 23.4 to illustrate how to tar-

get an interest rate risk exposure that differs from zero.

Example 23.5:Using DV01s to Form Imperfect Hedges

The 7-year bond from the last example has a DV01of $0.05 per $100 face value.Assum-

ing that changes in the yields of bonds of various maturities are identical, how much of the

7-year bond should be bought to change the corporate risk exposure, measured currently

as a DV01of $1,400, to a risk exposure with a DV01of $400?

Answer:Because portfolio DV01s are additive, if we buy $xface amount of the 7-year

bond, the DV01of the “portfolio”consisting of the corporation and $xof the 7-year bonds is

.05x

$1,400

100

The sum of these two terms is $400 when xequals $2 million.Thus, the $2 million face

amount of 7-year bonds generates the target interest rate risk exposure.

How Compounding Frequency Affects the Stated DV01

The stated yield to maturity of a bond depends on the compounding frequency used

for the yield.⁸Hence, the meaning of a 1 bp decline in yield and the size of the cor-

responding DV01depend on whether annual, semiannual, or monthly compounding is

used. The customary frequency for reporting DV01s and yields to maturity depends on

the bond’s coupon frequency. For example, DV01s for residential mortgages are typi-

cally based on monthly compounded 1 bp declines. DV01s for corporate bonds are

reported from semiannually compounded 1 bp declines.

⁸See Chapter 9.

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Chapter 23

Interest Rate Risk Management

825

For hedging purposes, the compounding frequency for the 1 bp decline is irrele-

vant as long as the analyst consistently uses the same compounding frequency when

computing DV01s for all the relevant investments. Maintaining this consistency might

require some DV01conversions. The DV01conversions between compounding fre-

quencies are given in the following result:

Result	23.2	Let rand rdenote the annualized yield to maturity of the same bond computed with yields
		nm
		compounded ntimes a year and mtimes a year, respectively. The DV01for a bond (or
		rr
		portfolio) using compounding of mtimes a year is nmtimes the DV01
		11
		nm
		ntimes a year.9
		of the bond using a compounding frequency of

9

Here is a calculus proof of Result 23.2: Equation (23.1) states that

dP

DV01 (for r) .0001

^mdr

m

while

dP

DV01 (for r) .0001

ⁿdr

n

where rand rare equivalent annualized rates of interest for compounding that occurs mtimes a year

mn

and ntimes a year, respectively. Dividing each side of the second equation into the corresponding sides

of the first equation, we get

DV01 (for r)dr

mⁿ

DV01 (for r)dr

nm

or

dr

DV01 (for r) ⁿ DV01 (for r)

^mdrⁿ

m

r

n

1

drn

To prove that ⁿ

,we refer the reader to Chapter 9. There, we learned that two equivalent

drr

^mm

1

m

rates that compound mtimes a year and ntimes a year satisfy the condition that the future value of a

dollar after one year must be the same; that is

rⁿr^m

n m

11

nm

After taking the natural logarithm of both sides of this equation, this is equivalent to

rr

n ln n m lnm

11

nm

Taking the derivative of both sides with respect to ryields

m

1dr1

rⁿ

drr

n^mm

11

nm

When rearranged, this says

r

n

1

drn

ⁿ

drr

^mm

1

m

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Strategy, Second Edition

826Part VIRisk Management