Duration

book 09 - Duration [oct 19] -- Page 1 of 18

Note for M-BIF students. You can skip the sections and subsections marked (*). We shall

cover these more advanced topics in the “Fixed Income” elective.

Chapter 9

Duration Metrics

Executive summary

Duration was the first quant-like risk metric widely adopted by Wall Street (in the early

1970s). It quickly became very popular, and, in its modern interpretation, is still the main

tool for managing interest-rate risk. To keep things straightforward, this first chapter on

duration will consider only plain vanilla fixed-interest bonds, and not securities

characterized by:

Variable cash flows, such as: floating rate notes or loans, interest rate swaps, inverse

floaters. Some of these topics will be covered in chapter 10.

Embedded options that entail the possibility of variable cash flows. For example,

callable and puttable bonds. These products will be covered in Part IV.

Convertible bonds (covered in Part IV).

The concept of duration was introduced by Macaulay (1937), with reference to straight

coupon bonds. However, we shall first consider the case of a zero coupon bond and then

extend the result to coupon bonds with the usual decomposition in a basket of zeros (see

chapter 7). This allows a much clearer understanding, and paves the way to the modern

concept of effective duration. We shall then examine the problems deriving from non-

parallel yield curve shifts and from the higher volatility of short-term yields. The chapter

concludes with an analysis of the key rates duration approach, first proposed by T. Ho

(1992).

Contents 9.1 - Duration metrics for discount securities

9.2 - Convexity

9.3 - Duration for coupon bonds: YTM and spot yield curve metrics

9.4 - The additivity of duration – handle with care

9.5 - Key rates duration


9.1 Duration metrics for discount securities

9.1.1 – Macaulay’s duration

The readers of this book are certainly familiar with Macaulay’s duration. Therefore, we

shall spend only a few words on this concept – not only out of historical interest, but also

because it is still used in a number of books and technical documents. Macaulay’s

duration is simply the weighted-average term of a bond’s cash flow, where the weights are

the present values of the individual payments, computed using the bond’s yield to maturity

(the weights add up to the bond’s invoice price). For a zero coupon bond, Macaulay’s

duration clearly equals the tenor of the zero. The following is the equation for Macaulay’s

duration, computed at the start of an interest-accrual period for an annual coupon bond

with (n) years to maturity.

Macaulay’s duration

[ ][ ]

(1) 2 (2) .. ( ) ( )(1) (2) .. ( ) ( )d

C d d n d n n F d nM

C d d d n F d n+ × + + × + × ×

=+ + + + ×

C = Coupon

( )d t =Discount factors

F =Face value (principal)

dM =Macaulay’s duration

Cash 6.00% Present Macaulay's

year flow d(t) values weights duration1 5.00 0.9434 4.7170 0.0503 0.0503 2 5.00 0.8900 4.4500 0.0474 0.0949 3 5.00 0.8396 4.1981 0.0448 0.1343 4 5.00 0.7921 3.9605 0.0422 0.1689 5 5.00 0.7473 3.7363 0.0398 0.1992 6 5.00 0.7050 3.5248 0.0376 0.2255 7 5.00 0.6651 3.3253 0.0355 0.2482 8 105.00 0.6274 65.8783 0.7024 5.6192

140.00 93.7902 1.0000 6.7404

Exhibit 9.1 – Computing Macaulay's duration for an 8-year 5.00% coupon bond, priced at

93.7902, to yield 6.00%.

Why did Macaulay call “duration” this average weighted maturity? Well, the expression

‘average maturity’ was already widely used to indicate the average life of a sinking fund

bond: Macaulay used the term duration simply not to ingenerate ambiguity. Note: The

bond’s cash flow comprises coupons and principal. Therefore, duration is easily computed


for sinking fund bonds. It turns out that for most categories of bonds (but not all!)

Macaulay’s duration is what drives the bond’s sensitivity to yield changes.

9.1.2 – Duration, a linear risk metric

A very well known (and intuitively evident) theorem of calculus states that a well-behaved

(smooth and continuous) curve can be approximated, within a small interval, by its tangent

(a straight line). This is known as first-order approximation. If we approximate the price-

yield function with its tangent computed at a given value of (Y), we get a linear interest-

rate risk metric for fixed-rate securities (see Exhibit 9.1). Linear expressions are easy to

use because they depend only on one number and allow quick and rough comparisons

between securities. The following two statements are equivalent; however, the second is

more compact and easier to grasp, also in the rapid-fire pace of a fixed-income dealing

floor.

The price of Bond A will decrease by 0.90% if market yield will increase by 10 basis

points (0.9% = 9 × 0.1%), while the price of bond B will decrease by 0.30% (0.3% =3 ×

0.1%). Therefore, Bond A is 3 times more interest-rate sensitive than bond B.

Bond A has duration of 9 versus 3 for bond B.

$20

$40

$60

$80

2% 4% 6% 8% 10%

Price

Tangent at Y = 0.06

Exhibit 9.2 – Linear approximation, computed at Y = 6.00%, to the price of a 12-year zero

coupon bond, face value $100

9.1.3 - Dollar duration

The tangent to the price-yield function for a zero is computed using the function of a

function rule, which can be found in all introductory calculus textbooks. The principal or


face value (F) is usually set at $1 or $100. Dollar duration is clearly a linear function of the

face value of the bond and of time-to-maturity (T). If you multiply the dollar duration by a

small variation in yield ( Y∆ ), you get the first-order approximation of the true dollar

variation in price ( P∆ ). Note that the name dollar duration is justified by the fact that

(dP/dY) is expressed in the monetary units of the bond price.

Dollar duration of a zero coupon bond

$

$

$

(1 )

1

TP F YdP PD TdY Y

P D Y

D

∆ ∆

−= +

= =−+

≈

= dollar duration

9.1.4 - The duration number

The derivative of the price/yield function is negative (for most bonds, such as zeros and

straight coupon bonds). Therefore, duration is a negative number. However, in the

securities industry, duration is often quoted as an absolute number (assuming that

professionals know its sign when carrying out their computations).

Example 9.1 – Consider a 12-year zero coupon bond (face value = $100), priced to yield

6% per annum. Let us compute, using the dollar duration approximation, by how much the

price would decline if the yield jumped to 6.20% (∆Y = 0.002)

12

$

$100(1.06) $49.6969

49.696912 $562.60681.06

$562.6068 0.002 -$1.12521

P

dPdY

P D Y

−= =

= − = −

∆ = ∆ = − × =

The $1.12521 drop in price computed with the linear duration approximation is rather

accurate. The true price drop is $1.1115. Due to the convexity of the price-yield function,

duration overestimates the loss due to an increase in (Y), and underestimates the gain

when (Y) decreases. This is sometimes referred to as gain from convexity (see section

9.2).


9.1.5 – Dollar duration with US bond yield quotation

When yield is quoted as twice the semiannual yield, the duration equation must be

modified accordingly.

Dollar duration using U.S. yield quotation

( )

2

$

12

1 2

TYP F

dP PD TYdY

−⎛ ⎞= +⎜ ⎟⎝ ⎠

= = −+

9.1.6 – Dollar value of one basis point (DV01)

Quite often dollar duration is expressed with reference to a 1 basis point yield variation.

This is known as price value of one basis point (PV01, or PVbp). Note, this measure is

most often denoted with DV01, dollar value of one basis point. Using PV01 is

straightforward. First compute its value, and then multiply it by the number of basis point

of the yield variation you are considering. PV01 is expressed with reference to a bond’s

face value.

Example 9.2 – Consider again a 12-year zero coupon bond (face value = $100), priced to

yield 6% per annum. Compute, using the PV01 approximation, by how much the price

would decline if the yield jumped to 6.20% (∆Y = 0.002). The result is clearly identical to

that obtained in example 9.1.

$

$

49.696912 562.60681.06

01 0.0001 0.05626068

01 20 $1.12521

dPDdY

PV D

PV

= = − = −

= × =

× = −

The DV01 point of a long-dated zero coupon bond first increases as a function of term to

maturity, and then declines, as shown in exhibit 9.2. This is due to two counteracting

factors:

The price-elasticity increases with maturity; therefore the DV01 increases rapidly as

the term increases.

Given a face value ($100 in exhibit 9.2), the price of the zero coupon bond declines

with the increase in tenor, and this produces a price-compression of dollar duration

and DV01.


$0.00

$0.01

$0.02

$0.03

$0.04

$0.05

$0.06

$0.07

0 5 10 15 20 25 30Tenor

DV01 for $100 zero

Exhibit 9.3 – DV01 as a function of time to maturity (tenor), for a $100 zero coupon bond,

computed for Y = 8.00%.

9.1.7 - Duration (also known as modified duration) for zero coupon bonds

Duration is the proportional variation in price that would be caused by a given absolute

variation in yield – to compute it, we simply divide dollar duration by price. This is

sometimes called modified duration because Macaulay’s duration is divided by (1+Y). In

current market practice, modified duration is called duration, tout court. As we can see

from the equation, the duration of a zero coupon bond is linearly proportional to (T). The

price-compression effect cancels out as (P) appears both at the numerator and the de-

nominator of the equation.

Duration (modified duration)

$ 1 1(1 ) (1 )

D dP P TD TP dY P Y P Y

= = = − = −+ +

Example 9.3 – Consider once more a 12 year zero coupon bond (face value = 100),

priced to yield 6% per annum. Use modified duration to compute by how much the price

would decline if the yield jumped to 6.20% (∆Y = 20 bp). As expected, you get the same

result of Examples 9.1 & 9.2

12 11.32081.06

$49.6969 11.3208 ( 0.002) $1.12521

D

DP P Y∆ ∆×

=− =

= ×

= × × − =−


9.1.8 – Endnotes

MODIFIED DURATION IS A PURE NUMBER (without units of measure) and is not expressed in

years. The statement “this bond has duration of 7.3 years” is inaccurate. Speaking in

terms of years is right for Macaulay’s duration, which is a measure of average life, but not

for modified duration, which is the proportional change in value of a bond due to an

absolute change in yield. This is easy to see when we consider that duration is computed

dividing dollar duration (expressed in dollars) by price, also expressed in dollars. For

example:

$ $562.6068 11.3208$49.6969

DD

P= = =

THE THREE DURATION METRICS (dollar duration, DV01, and modified duration) are linked by

the following equations:

$

$

01 0.0001

01/ 0.0001

DV D

D DVDP P

= ×

= =

USING LOG YIELDS, dollar duration and duration are given by the following equations: ( )

( )$

$

exp

exp

P F RT

dPD F RT TdRD

D TP

= × −

= = − × −

= = −

9.2 Convexity (*)

9.2.1 – Convexity and tenor

Convexity is measured by the second derivative of the price-yield function. Therefore, it

increases with the square of time to maturity. A good way to show convexity is to compare

the price-yield function, around a given value (Y*), with its tangent at the chosen point (Y*)

(exhibit 9.4).


Convexity of a zero

( )

( )

( 1)

2( 2)

2

2

( 2)

(1 )

1 (1 )

(1 )

T

T

T

dP T YdY

d P YdY

TT Y

T T

− +

− +

− +

=− +

= + +

= + +

For more complex bonds (for example, coupon bond with amortization schedule) the

derivative can be computed numerically.

-$2

$0

$2

$4

$6

$8

2% 4% 6% 8% 10%

Convexity error

Exhibit 9.4 – Convexity error of the duration approximation of the price of a zero coupon bond

(Face value = $100, tenor = 12 years, Y = 6.00 %.)

9.2.2 – Convexity approximation

A smooth function can be approximated using both the fist and the second derivatives

(second order approximation). This property can be used to write an equation to

approximate the price-yield function in terms of dollar duration and convexity (measured

by the 2nd derivative). Note: in options-speak, Delta, and Delta-Gamma denote

respectively the first and second order approximations.

The second order approximation is very accurate, but (contrary to duration) is not

much used in business life. Duration is a rough measure but has the advantage of being

expressed by one number. Convexity approximation is more accurate, but requires

computation. This is the equation, where (C) indicates convexity.


Convexity approximation 2

22

2$

1 ( )21 ( )2

dP d PP Y YdY dY

D Y C Y

∆ ∆ ∆

∆ ∆

= +

= +

-$2

$0

$2

$4

$6

$8

2% 4% 6% 8% 10%

Duration approx

Duration-Convexity approx

Exhibit 9.5 – Duration and convexity approximation to the price of a zero coupon bond (face

value = $100, tenor = 12 years). The yield at which approximation is computed = 6.00%. In

order to make the figure easily readable, the approximations are measured by their errors

relative to the true price of the zero coupon bond.

9.3 Duration of coupon bonds: YTM and spot yield curve metrics

9.3.1 – Duration of a coupon bond

The original formulation of duration was in terms of yield to maturity for straight coupon

bonds – in those days, YTM was the only well known yield measure. It’s intuitively evident

that the dollar duration and the DV01 of a coupon bond equal respectively the sum of the

dollar durations and of the DV01 of the set of zeros into which the bond can be

decomposed. This intuition is supported by a well-known theorem of calculus that

establishes that the derivative of a sum of functions equals the sum of their derivatives.

If we use YTM, we must assume a flat yield curve, at a level equal to the yield to

maturity of the coupon bond. Consider exhibit 9.6. Each of the eight dollar durations is


computed using the 6.00% YTM. For example, the dollar duration of the last payment

(which is inclusive of principal) is given by the following equation:

8105 1.06497.19 81.06

−×− =−

Cash PV at Dollar

year flow YTM = 6.00% duration DV01

1 5.00 4.7170 -4.45 -0.0004

2 5.00 4.4500 -8.40 -0.0008

3 5.00 4.1981 -11.88 -0.0012

4 5.00 3.9605 -14.95 -0.0015

5 5.00 3.7363 -17.62 -0.0018

6 5.00 3.5248 -19.95 -0.0020

7 5.00 3.3253 -21.96 -0.0022

8 105.00 65.8783 -497.19 -0.0497

140.00 93.7902 -596.40 -0.0596

Exhibit 9.6 – Computing dollar duration and DV01 for an 8-year 5.00% coupon bond, priced to

yield 6.00%.

We can also prove that for straight bonds (and for fixed-interest amortizing bonds)

modified duration equals Macaulay’s duration divided by (1+YTM).

1DMDYTM

=+

9.3.2 - Parallel shift duration for coupon bonds (Fisher-Weil duration) *

A more recent interpretation of duration is based on the idea of considering the effect of a

small parallel shift on the spot yield curve. This approach is clearly more in line with the

modern SYC-based approach, and with VaR methods (see Part V). However, from a

practical point of view the numerical difference is not meaningful, as we can see

considering the Exhibit 9.7, where we compute the effect on a bullet bond price of a +1bp

change in YTM, and of a +1bp spot yield curve parallel shift. We can see the resulting

prices are very close.

When we use the SYC parallel shift method, we obtain directly the DV01. From here

it’s easy to compute duration (modified duration). In the following equation we use the ∆P

value obtained with a 1bp upwards shift of the spot yield curve.


1 0.0595 1 6.339293.7904 0.0001

P D P Y

PDP Y

∆ = × × ∆

∆= = − ⋅ = −

∆

CPN & Discount Factors

years Pricipal SYC SYC SYC+ 1bp YTM + 1bp

1 5.00 1.8112% 0.9822 0.9821 0.9433

2 5.00 2.6912% 0.9483 0.9481 0.8898

3 5.00 3.4912% 0.9022 0.9019 0.8394

4 5.00 4.2112% 0.8479 0.8476 0.7918

5 5.00 4.8512% 0.7891 0.7887 0.7469

6 5.00 5.4112% 0.7289 0.7285 0.7046

7 5.00 5.8912% 0.6699 0.6694 0.6646

8 105.00 6.2912% 0.6138 0.6133 0.6269

Price = 93.7902 93.7312 93.7306

DV01 = -0.0590 -0.0596

Duration -6.2900 -6.3564

Exhibit 9.7 – SYC and YTM duration for an 8-year, 5.00% coupon bond, priced at 93.7902, to

yield 6.00%. Modified durations are computed numerically, based on the DV01s.

.

9.4 The additivity of duration – handle with care (*) We have used the additivity of duration to compute the duration of a coupon bond – in the

same way; we can utilize the additivity of duration to summarize in one number the

duration of a portfolio of fixed income securities. This is indeed a useful characteristic,

provided that it is used mainly for financial accounting and reporting purposes, such as:

“The overall duration of our portfolio is 5.2.” The big problem with duration hedging is that

it postulates parallel yield curve shifts. This implies that – using duration acritically – you

could be tempted to hedge a 20-year bond with the equivalent duration-weighted amount

of one-year bonds (or 1 month bills if you prefer). This rather surprising hedge will, in fact,

work quite well for small parallel shifts of the yield curve, as shown in exhibit 9.8.


Maturity 1 10 Spot yield 2.00% 6.00%

Price 98.039216 55.839478DV01 -0.009611 -0.052651

Hedged principals 5,478,392 1,000,000DV01 (on principals) -526.51 -526.51

Exhibit 9.8 – Hedging a 10-year zero coupon bond with a one year zero, assuming parallel

yield curve shifts. The principal of the long zero is set at $1million, while the principal of the one

year zero is computed with the ratio of durations. Source of data: Federal Reserve website

(see in references).

The problems with this naïve use of duration are:

Most of the time yield curves do not move in a parallel way

The short-term end of the yield curve usually exhibits a higher variability than the long-

term end (see exhibit 9.9).

For largish yield curve shifts, convexity plays its role, and long bonds are a lot more

convex than short bonds. (This higher convexity underlies trading strategies, such as

barbell trades. See the "Additional Notes PDF file" for this chapter.)

0

1

2

3

4

5

6

7

Jan-99 Jan-00 Jan-01 Jan-02 Jan-03

3 Month bills 10 year note

Exhibit 9.9 – U.S. Treasury rates. It is easy to see that the short term rate displays much wider

swings than the 10-year yield. In 1999 and 2000, the FED’s policy was mildly restrictive.

Towards the end of 2000 the monetary policy stance changed dramatically to provide stimulus

to the economy that was sliding in recession due first to the crisis in the tech sector, and then to

the Sept. 11 terrorist attacks.

A number of duration (and convexity) strategies are seriously flawed because they do not

take into account probable non-parallel yield curve shifts. On the other hand, in some spe-


cial cases, duration-based approaches can be useful. One of them is the hedging of a

bond (swap) position (long or short) with a benchmark, on-the-run bond

Benchmark bonds are more liquid than the average and can be traded in volume with

lower transaction costs. They are also more available for borrowing to set up short posi-

tions. Assume you are a syndicate desk manager or a block trader, and that you have a

position in 8-year AAA dollar bonds. Now you can well hedge this position (short-lived)

with a 10-year on the run Treasury note, instead of trying to do it with an 8-year note. A

quick glance at the Exhibit 9.10 shows that rate correlation is very high in that horizon

range. However, this level of correlation is hardly sufficient if you want to set up leveraged

positions for relative value strategies (see the Long Term Capital Management case).

-0.6%

-0.4%

-0.2%

0.0%

0.2%

0.4%

0.6%

Jan-99 Jan-00 Jan-01 Jan-02 Jan-03

Yield Dif ferential: 10Y - 7Y

Exhibit 9.10 – Yield spreads between the 10-year and the 7-year U.S. treasury notes. Source

of data: Federal Reserve website (see in references).

Of course your hedge must be duration-adjusted, and this is very straightforward,

especially if you use dollar duration, in the shape of DV01.

01( )( ) ( )01( )

DV AAAF T F AAADV T

=

Where F(T) and F(AAA) indicate respectively the face value of the treasuries and of the

AAA-rated corporate bonds and DV01 is computed at the current level of yields. The

above hedge would work even better if you were hedging treasuries with treasuries. The

problem with our AAA case is that spread over treasuries can change rapidly over time

(this was one of the problems which provoked the collapse of Long Term Capital

Management LL.PP.).


9.5 Key rates duration (*) We have seen that duration is heavily dependent on unrealistic assumptions about near-

parallel yield curve shifts. An interesting (and easy to implement) way to overcome the

limitations of traditional duration was proposed by T. Ho (1992). This approach consists is

starting with the spot yield curve and defining a certain number of key maturities, aka

factors. T. Ho proposed using a key maturity for each of the on-the-run treasuries, but we

shall simplify our description by using only 3 key rates:

{2-year T-note, 10year T-note, 20year T-bond}

The next step is based on the simplifying assumption that a change in any key rate will

not, by itself, change the adjacent key rates but will change the intermediate rates in a

linearly declining fashion. This sounds like a mouthful but is easily visualized in exhibit

9.11 that shows the effect of a 100 basis points increase of the 10-year key rate. For

example, on the left-hand side, the next key rate is that of the 2-year T-note, and this will

not be affected by the change in the 10-year rate. Thus, the effect of the 100 basis point

increase will decline by 1/8th for every year of tenor. The effect on the 5-year rate will be:

( )58100 1 0.375%− =bp

0

20

40

60

80

100

120

0 5 10 15 20 25

Tenor

Bas

is p

oint

s

Exhibit 9.11 – Effect of a 100 basis point change of the 10-year key rate.

If you assume that there is more than one key rate change the effect will simply be that of

adding together the linear changes, as shown in exhibit 9.12.


0

20

40

60

80

100

120

140

0 5 10 15 20

Tenor

Bas

is p

oint

s

Exhibit 9.12 – Changes in all key rates. Two year key rate +120 bp; Ten year key rate +100

bp; Twenty year key rate +80bp.

Given this construction you can define the duration of a fixed-income instrument with

respect to each key rate. A 7year note will clearly have duration relative to the short-term,

2year and 10year key rates. And its total duration will be the sum of the relevant key rate

durations.

Summary and Conclusions In this chapter we have covered a number of topics that cover the modern interpretation of

duration:

Duration as a linear risk measure. It can be measured with three interconnected

metrics: dollar duration (used in first order approximation), dollar value of one basis

point (DV01), and modified duration.

Duration and parallel shifts in the spot yield curve (Fisher-Weil duration, commonly

known as effective duration). This approach is functional to the analysis of fixed

income derivatives and interest-rate risk management.

Convexity and convexity approximation.

Duration of coupon bonds – showing how the traditional yield-to-maturity based

duration is a special case of the parallel spot yield curve shift duration (Fisher-Weil

duration).


The additivity of duration. This property is clearly very useful (we can compute the

duration of a bond portfolio). It is also inherently dangerous in risk management

applications because yield curve shifts are most often non-parallel.

The problems caused by non-parallel yield curve shifts for duration hedging.

Key Terms Convexity

Convexity Error

Dollar Duration

Dollar Value of One Basis Point (DV01)

Duration

Effective duration

First-order approximation

Invoice price

Macaulay’s dutation

Modified Duration

Monotonic Curve

Price-Compression

Price Value of one Basis Point (PV01)

Second-order approximation

Straight Coupon Bonds

Tangent

Taylor Series


References and suggested readings

Corporate finance and investments textbooks

Benninga, S. Z., and B. Czaczkes (2000). Financial Modeling, 2nd Edition. MIT Press

Bodie Z., A. Kane, and A. J. Marcus (2002) Investments., 5th Edition. New York: McGraw-

Hill/Irwin

Grinblatt, M. and S. Titman (2002): Financial Markets and Corporate Strategy, 2nd Edition.

New York: McGraw-Hill/Irwin.

Fixed income and risk management books

Fabozzi, F. J. (2000). The Handbook of Fixed Income Securities, 6th Edition, McGraw-Hill

Professional

Jorion, P. (1995). Big Bets Gone Bad: Derivatives and Bankruptcy in Orange County,

Academic Press, San Diego, Calif.

Jorion, Philippe (2001). Value at Risk, 2nd Edition. New York: McGraw-Hill

Macaulay, F.R. (1938). Some Theoretical Problems Suggested by the Movements of Interest

Rates, Bond Yields and Stock Prices in the US Since 1856, National Bureau of Economic

Research, New York

Questa, Giorgio S. (1999). Fixed Income Analysis. John Wiley & Sons, New York

Sundaresan, Suresh M. (2001). Fixed Income Markets and Their Derivatives, 2nd Edition,

Dryden Press

Tuckman, Bruce (2002). Fixed Income Securities, 2nd Edition, John Wiley & Sons, New York.

Journal articles and case studies

Fisher, Lawrence, and Roman L. Weil (1971), “Coping with the Risk of Interest-Rate Fluc-

tuations: Returns to Bondholders from Naïve and Optimal Strategies.” The Journal of

Business, Volume 44, Issue 4 (Oct., 1971) 408-431

Ho, Thomas (1992). “Key Rate Durations: Measures of Interest Rate Risk.” The Journal of

Fixed Income (Sep., 1992) 29-43

Lewis, Michael (1999), ‘How the eggheads cracked’, New York Times Magazine, January 24,

pp. 24-77.

Perauld, A. F. (1999), “Long-Term Capital Management L.P.” Harvard Business School case

studies [A, B, C, D].

Schaefer, Stephen M. (1984). "Immunisation and Duration: A Review of Theory, Performance

and Applications.", Midland Corporate Finance Journal, Vol. 2, pp 41-58


Documents and statistics

Basel Committee on Banking Supervision, Principles for the Management and Supervision of

Interest Rate Risk, Consultative Document, Basel 2001. (Available on www.bis.org)

The US Federal Reserve (http://www.federalreserve.gov/releases/h15/data.htm#fn3)