Immunization and Large Changes in Interest Rates

A$10,000 6-year 8 percent straight-coupon bond with annual coupons trading at par

has a duration of approximately five years. If the term structure of interest rates is flat,

and the interest rate increases slightly, the reinvested coupons will have an increased

value at year 5 that exactly counterbalances the decreased value of the bond in year 5

(that is, the year 5 value of its cash flows after year 5). The same is true when there

is a small decline in interest rates.

It is important, however, to determine how well immunization strategies perform

when there are substantial interest rate changes. Interest rates can sometimes jump by

large amounts when important economic indicators are announced (for example, the

monthly employment report by the U.S. Department of Labor or the Federal Reserve’s

announcement of its target interest rate). Moreover, even a series of small changes in

interest rates can amount to a large effective change in interest rates if transaction costs

lead investors to undertake a lax immunization strategy—failing to maintain the proper

duration for the bond at all points in time.

One can be comforted that immunization strategies work “pretty well” for the 8

percent straight-coupon bond described above and even for large interest rate changes

such as 25 basis points in one day. A25 bp decline in interest rates, from 8 percent to

7.75 percent, makes the sum of the year 5 value of the bond and its reinvested coupons

equal $14,693.14; in the absence of an interest rate shift, the bond’s value is $14,693.28.

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Chapter 23

Interest Rate Risk Management

839

This is a remarkably insignificant difference! Since a 25 bp shift in the yield of a 6-year

bond would rarely occur over a single day, we can be fairly confident in stating that

immunization in this case would not require exceptionally frequent rebalancing of the

portfolio.

The next section discusses how to better quantify this degree of confidence.

23.5Convexity

Convexity measures how much DV01changes as the yield of a bond or bond portfo-

lio changes. Aportfolio with a DV01of zero will be insensitive to small interest rate

movements and less sensitive to large interest rate movements the smaller the convex-

ity. In other words, the DV01remains close to zero as interest rates change if convex-

ity is close to zero.

Defining and Interpreting Convexity

Convexity is a measure that determines how secure an investor should feel about a

“perfectly hedged” portfolio (for example, whether an electronic feed of bond prices

requires constant monitoring or whether the investor can relax and do other things). If

the convexity of a hedged portfolio is zero, even fairly large changes in interest rates

over a short time span should not alter the value of the portfolio drastically. If the con-

vexity is large, constant monitoring of the bond prices might be a good idea.

Positive Convexity Is More Typical.Most bonds have positive convexity, which

expresses the type of curvature seen earlier in the price-yield curve (Exhibit 23.1).

Exhibit 23.6 portrays price-yield curves for two bonds: one with a large amount of

convexity and one with little convexity. In contrast to this picture, bond portfolios,

withlong and short positions in different bonds, can have positive or “negative

convexity.”¹⁵

EXHIBIT23.6Convexity

Price	Price
Large amount of convexity	Little convexity

Yield

Yield

¹⁵Despite

its applications in science and mathematics, the word concavity,which is probably a better

term than negative convexity,has yet to find a place in the vocabulary of the bond markets.

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840Part VIRisk Management

The Convexity Formula.At a single point on the price-yield curve for a bond or

bond portfolio, the curvature, measured by convexity, is formally defined as follows:

The convexity “CONV(r)” of a bond or a bond portfolio at a yield to maturity of ris

the product of (1) $1,000,000 divided by the bond price and (2) the difference between

the current DV01(of the entire bond position) and the DV01(of the entire bond posi-

tion) at a yield of r.0001; that is

CONV(r) ^$1 million

DV01(r)DV01(r.0001)

P

DV01can be viewed as a derivative of the bond price with respect to its yield to

maturity. It differs from the derivative primarily in that DV01multiplies the derivative

by a constant. Convexity, CONV,is like the second derivative of the portfolio’s price

with respect to its yield in percent (not its basis point shift). To obtain convexity, mul-

tiply the DV01differenceby 10,000 (the square of 100) to undo the DV01convention

of multiplying the derivative with respect to the percentage yield by .01 (100 bps is

1percentage point of interest). In addition, convexity is typically scaled per $100 of

market value. Hence, after multiplying the difference between the DV01s at the two

yields by 10,000, it is customary to multiply by $100 and divide by the bond price per

$100 of face value. Example 23.14 illustrates the calculation.

Example 23.14:Computing Convexity

Compute the convexity of the 8 percent 2-year par straight-coupon bond analyzed in Exam-

ple 23.6.

Answer:The DV01of this bond is .0181516 per $100 face value at an 8.00 percent yield,

since its price at 7.99 percent is $100.0181516.At an 8.01 percent yield, the DV01is

.0181473, the difference between the $100 price at 8.00 percent and the price at 8.01 per-

cent, which is $99.9818527.The difference between the two DV01s is 4.3 10⁶.The

con-

vexity is .043, which is 1 million divided by 100 times this number.

Implications of Convexity forImmunization Strategies.The last section indicated

that immunizing a portfolio for a T-year horizon requires managing the portfolio, so

that when it is combined with a hypothetical short position in a zero-coupon bond of

Tyears maturity, the duration and value of the combined portfolio is zero. If the con-

vexity of the actualportfolio is close to the convexity of the hypothetical T-year zero-

coupon bond, the investor who is immunizing the portfolio does not have to constantly

monitor bond prices. However, frequent rebalancing is needed to maintain the immu-

nized position of the actual portfolio if its convexity differs substantially from that of

the hypothetical portfolio. In the latter case, constant bond price monitoring is required.

Estimating Price Sensitivity to Yield

Investors often use convexity to improve upon the earlier (DV01or duration-based)

estimate of the change in the price of a bond portfolio for a given change in yield.

Specifically, letting rdenote the change in the yield to maturity (in percent form) and

Pthe starting price of the bond or bond portfolio, we can write¹⁶

¹⁶This

equation is derived from the second order Taylor series expansion. See any elementary calculus

text for details.

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Chapter 23

Interest Rate Risk Management

841

P
CONV(r)2
P100DV01 r.5	(23.8)
100

Example 23.15 illustrates how to apply the formula.

Example 23.15:Using Convexity forAccurate Estimates of Price Change

Compute the change in the price of a bond portfolio with a $200 market value for a 25 basis

point increase in yield.At the current yield to maturity, the bond has a DV01of $0.15 and

a convexity of 1.2 (convexity is computed per $100 of market value).

Answer:Using equation (23.8), the change in price is

$200

2

100($.15)(.25).5 (1.2)(.25) $3.675

100

Hence, the new price will be $196.325.

One would estimate the change in price in Example 23.15 as $3.75 using only

DV01to estimate the change. Using equation (23.8) thus improves the estimate of inter-

est rate sensitivity by $.075.

Convexity (unlike DV01,but like duration) is independent of the scale of the invest-

ment. Holding twice as many bonds of the same type does not change the convexity

of the portfolio. Moreover, the convexity of a portfolio of bonds is the value-weighted

average of the convexity of each bond in the portfolio.

Misuse of Convexity

Throughout this chapter, we have assumed that the term structure of interest rates is

flat. This is the traditional approach taken in all but the most sophisticated bond port-

folio analysis and it is a good way to begin to understand the issues in bond portfolio

management and interest rate hedging. However, we must express caution: Not only is

this assumption generally an incorrect portrait of realistic term structures, but it is

fraught with a number of pitfalls that are based on inherent flaws in logic. Amisguided

application of convexity, discussed next, illuminates this point.

Convexity Appears to Be Good.Equation (23.8) implies that the higher the con-

vexity of a bond portfolio (holding market value and DV01constant), the larger the

value of the portfolio for both an increase and a decline in its yield to maturity. This

seems to imply that convexity is a good thing to have in a portfolio and that, other

things being equal, investments with a large amount of convexity are in some sense

better than investments with little convexity or negative convexity. Exhibit 23.7 illus-

trates this point by plotting the price-yield curve for two default-free bonds, each with

the same market value and DV01at a yield of 8 percent. Note that as yields change,

irrespective of the direction of change, the price of bond Awill be higher than that of

bond B. Bond Athus appears to be the better bond because it has more convexity.

The flaw in this analysis is that the yields to maturity of the two bonds need not

be the same. Unless the term structure of interest rates is flat, the two bonds need not

have the same yields at the same time, and when one bond’s yield moves up, the other’s

yield need not move up by the same amount. Comparing yields between two bonds is

a comparison of apples and oranges.

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842	Part VIRisk Management EXHIBIT23.7The Effects of Convexity forSimilarBonds Price

100
	Bond A

Bond B

	Yield
8%

Arbitrage Exists When the Term Structure of Interest Rates Is Always Flat.

What if the term structure of interest rates is always flat? Is it then possible to con-

struct a pair of bond portfolio investments that look like bond Aand bond B? Or is

the construction of two such investments impossible? In the flat term structure sce-

nario, it is always possible to create two portfolios with the same market value and

the same DV01,but with different convexities. (They already have the same yield to

maturity by virtue of the assumption of a flat term structure of interest rates.) How-

ever, if one could find two bond portfolios with the characteristics of bonds Aand B

in Exhibit 23.7 there would be an arbitrage opportunity. By going long in the high-

convexity portfolio and short in the low-convexity portfolio, one achieves an invest-

ment combination that is self-financing and—for any size move in the interest rate—

has a positive value immediately after such move. What this means is that the

relationships depicted in Exhibit 23.7 are unlikely to exist in reality.

Changing the Convexity of a Bond Portfolio.An easy way to increase convexity

while holding DV01and market value constant is tospread payments out around

theduration.¹⁷

For example, compare (1) a portfolio consisting of a single 4-year

zero-coupon bond worth $2 million, with (2) a $2 million bond portfolio with equal

¹⁷Note

also that bond options, whether put or call options, have more convexity than the underlying

bonds themselves. Similarly, floating-rate investments with caps and floors have more convexity than

otherwise identical floating-rate investments without caps or floors.

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Strategy, Second Edition

Chapter 23

Interest Rate Risk Management

843

amounts invested in a 3-year zero-coupon bond and a 5-year zero-coupon bond. With

a flat term structure curve, the duration of the 2-bond portfolio is the value-weighted

average of the durations of the 3- and 5-year bonds, or four years—just like the first

zero-coupon bond. The DV01of the 2-bond portfolio is virtually identical to the DV01

of the 1-bond portfolio¹⁸because the durations and market values of the two portfo-

lios are the same.

If interest rates decline by 1 bp, the initial effect on the values of the two portfo-

lios will be about the same. However, since the 5-year bond in the 2-bond portfolio

now carries relatively more weight, the duration of the 2-bond portfolio will now be

closer to five years than to three. For the next basis point decline in interest rates, the

price of the 2-bond portfolio, which has a duration exceeding four years, will go up by

more than the price of the 1-bond portfolio, which has a duration of exactly four years.

This would push the duration even closer to five years. With a higher price and a still

higher duration, the DV01of the 2-bond portfolio would then exceed the DV01of the

1-bond portfolio by even more, and so on.

Consider the reverse situation. As interest rates increase, the duration of the 3-year

bond in the 2-bond portfolio now carries greater weight. Hence, while the first basis

point increase has about the same effect on both portfolios, subsequent basis point

increases result in greater price declines for the 1-bond portfolio because it has a larger

duration. In this case, as interest rates move down from the original rate, the DV01of

the 2-bond portfolio exceeds the DV01of the 1-bond portfolio by increasingly greater

amounts. Since DV01s are proportional to the slopes in a price-yield graph, the price-

yield graphs of the two portfolios look like those depicted in Exhibit 23.7 where the

1-bond portfolio is bond B and the 2-bond portfolio is bond A.

The previous discussion showed how to increase convexity by spreading payments

around the duration of a bond, focusing on zero-coupon bonds. It is possible to gener-

alize this procedure to coupon bonds. Panel Aof Exhibit 23.8 plots the cash flows of

an annuity bond against time. Panel B graphs the annuity bond’s price yield curve to

illustrate the convexity of the bond. Panel C plots the cash flows of a portfolio of two

zero-coupon bonds of different maturities. In panel D, the convexity of the 2-bond port-

folio is greater than the convexity of the annuity bond in panel B, seen as greater cur-

vature in the former. Example 23.16 uses numerical computations to demonstrate this

difference in convexity between an annuity bond and a portfolio of two zero-coupon

bonds.

Example 23.16:Convexity of an Annuity versus a Portfolio of

Zero-CouponBonds

Consider a portfolio consisting of an annuity that pays $100 every year for the next 30 years.

If 8 percent compounded annually is the market interest rate at all maturities, then the pres-

ent value of this annuity at 8 percent is

11

$1001 $1,125.7783

1.08³⁰

.08

Its present values at 7.99 percent and 8.01 percent interest rates are, respectively

11

$1001 $1,126.8413

.07991.0799³⁰

¹⁸The

derivative implementations of the two DV01s are exactly the same.

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844Part VIRisk Management

EXHIBIT23.8Spreading Cash Flow around the Duration of a Coupon Bond

Coupon

Panel A

Price

Panel B

Annunity

Bond

t

Yield

Coupon

Panel C

Price

Panel D

Portfolio of two

zero-coupon bonds,

1 has a short maturity date

1 has a long maturity date

t

Yield

and

11

$1001 $1,124.7170

1.0801³⁰

.0801

The DV01s at 8.01 percent and 8.00 percent interest thus equal $1.0613 ( $1,125.7783

$1,124.7170) and $1.0630 ( $1,126.8413$1,125.7783), respectively, implying that the

annuity bond’s convexity is

$1,000,000

(1.06301.0613) 1.510

$1,125.7783

Replacing this annuity with a portfolio consisting of zero-coupon bonds maturing at year 1

and year 30 can increase convexity while maintaining the same DV01at 8 percent interest.

Find such a portfolio.

Answer:The problem requires finding a portfolio with face amounts xand yin the 1-year

and 30-year zero-coupon bonds, respectively.For a $1.00 face amount, note that a 1-year

zero-coupon bond has a DV01of $.000085742, while a 30-year zero-coupon bond has a

DV01of $.000276445.Thus, to generate a portfolio with the same present value and DV01

as the annuity, xand ymust satisfy the equations

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Chapter 23

Interest Rate Risk Management

845

xy

$1,125.7783 (present value condition)

1.08(1.08)³⁰

.000085742x.000276445y $1.0630 (DV01 condition)

y

The first equation says x $1,125.7783(1.08).Substituting this into the second

(1.08)²⁹

equation gives

y

(.000085742)($1125.7783) (1.08).000276445y $1.0630

(1.08)²⁹

or	y(.000276445.000009202476147) $0.958747165213

Thus, y $3,587.60 and x $830.80 (approximately).

The DV01of the portfolio of zero-coupon bonds is 1.0602 at an 8.01 percent yield and

1.0630 at an 8.00 percent yield.This makes convexity

$1,000,000

(1.06301.0602), or 2.487

$1,125.7783

Example 23.16 illustrates how easy it is to increase convexity at no cost. Using equa-

tion (23.8), the difference in convexities for a 25 bp shift upward or downward (to

either 7.75 percent or 8.25 percent) implies that the portfolio of two zero-coupon bonds

has a larger price than the annuity by the difference in the final terms in equation (23.8):

$1125.778

.52

(2.4871.510)(.25) $.34

100

Hence, going long in the portfolio of zero-coupon bonds and short in the annuity gen-

erates approximately a riskless $.34 for a portfolio of this size.¹⁹Most portfolios would

be much larger, particularly since this is a hedged portfolio. At 10,000 times the size,

$3,400 is achieved without risk; at 100,000 times the size, $34,000 is achieved with-

out risk.

It is difficult to believe that such arbitrage profits can be achieved. However, if the

term structure of interest rates is always flat, there is no cost to forming a portfolio

strategy in this manner. It is easy to prove that arbitrage profits also arise if the yield

curve is not flat but only shifts in a parallel manner. If convexity is to come at a cost

and if there is no arbitrage, term structure changes cannot be restricted to parallel shifts.