Chapter 5

Hedging Interest-Rate Risk with Duration

FIXED-INCOME SECURITIES

Outline

• Pricing and Hedging– Pricing certain cash-flows– Interest rate risk– Hedging principles

• Duration-Based Hedging Techniques– Definition of duration– Properties of duration– Hedging with duration

Pricing and Hedging Motivation

• Fixed-income products can pay either– Fixed cash-flows (e.g., fixed-rate Treasury coupon bond)– Random cash-flows: depend on the future evolution of interest

rates (e.g., floating rate note) or other variables (prepayment rate on a mortgage pool)

• Objective for this chapter– Hedge the value of a portfolio of fixed cash-flows

• Valuation and hedging of random cash-flow is a somewhat more complex task– Leave it for later

Pricing and HedgingNotation

• B(t,T) : price at date t of a unit discount bond paying off $1 at date T (« discount factor »)

• Ra(t,) : zero coupon rate – or pure discount rate, – or yield-to-maturity on a zero-coupon bond with maturity date t +

θa θtR

θttB)),(1(

1),(

),(ln1),( θttBθ

θtR

),(exp),( θtRθθttB

• R(t,) : continuously compounded pure discount rate with maturity t + :

– Equivalently,

• The value at date t (Vt) of a bond paying cash-flows F(i) is given by:

105100100%55100%5

NcNFcNF

m

i

• Example: $100 bond with a 5% coupon

• Therefore, the value is a function of time and interest rates– Value changes as interest rates fluctuate

m

ii

a

im

ii

itRFittBFtV

11 ),(1),()(

Pricing and Hedging Pricing Certain Cash-Flows

• Example– Assume today a flat structure of interest rates– Ra(0,) = 10% for all – Bond with 10 years maturity, coupon rate = 10%– Price: $100

• If the term structure shifts up to 12% (parallel shift)– Bond price : $88.7 – Capital loss: $11.3, or 11.3%

• Implications– Hedging interest rate risk is economically important – Hedging interest rate risk is a complex task: 10 risk factors in this

example!

Pricing and Hedging Interest Rate Risk

• Basic principle: attempt to reduce as much as possible the dimensionality of the problem

• First step: duration hedging– Consider only one risk factor– Assume a flat yield curve– Assume only small changes in the risk factor

• Beyond duration– Relax the assumption of small interest rate changes– Relax the assumption of a flat yield curve– Relax the assumption of parallel shifts

Pricing and Hedging Hedging Principles

• Use a “proxy” for the term structure: the yield to maturity of the bond– It is an average of the whole terms structure– If the term structure is flat, it is the term structure

• We will study the sensitivity of the price of the bond to changes in yield:– Change in TS means change in yield

• Price of the bond: (actually y/2)

Duration HedgingDuration

m

ii

i

yFV

1 1

Duration Hedging Sensitivity

)()( yVdyyVdV

dyyVdV )('

dySensdyyVyV

VdV

)()('

• Interest rate risk– Rates change from y to y+dy– dy is a small variation, say 1 basis point (e.g., from 5% to 5.01%)

• Change in bond value dV following change in rate value dy

• For small changes, can be approximated by

• Relative variation

• The relative sensitivity, denoted as Sens, is the partial derivative of the bond price with respect to yield, divided by the bond price

• Formally

Duration Hedging Duration

/)(11

1

)()(' 1

yVy

iFy

yVyVSens

m

ii

i

• In plain English: tells you how much relative change in price follows a given small change in yield impact

• It is always a negative number– Bond price goes down when yield goes up

• The opposite of the sensitivity Sens is referred to as « Modified Duration »

• The absolute sensitivity V’(y) = Sens x V(y) is referred to as « $ Duration »

• Example: – Bond with 10 year maturity– Coupon rate: 6%– Quoted at 5% yield or equivalently $107.72 price– The $ Duration of this bond is -809.67 and the modified duration is

7.52.

• Interpretation– Rate goes up by 0.1% (10 basis points)– Absolute P&L: -809.67x.0.1% = -$0.80967– Relative P&L: -7.52x0.1% = -0.752%

Duration Hedging Terminology

• Definition of Duration D:

• Also known as “Macaulay duration”• It is a measure of average maturity

• Relationship with sensitivity and modified duration:

m

i

ii

VyFi

D1

)1(

Duration Hedging Duration

)1()1( yMDySensD

Time of Cash Flow (i)

Cash Flow

F i i

ii y

Fw

1V

1 iwi

1 53.4 0.0506930 0.0506930

2 53.4 0.0481232 0.0962464

3 53.4 0.0456837 0.1370511

4 53.4 0.0433679 0.1734714

5 53.4 0.0411694 0.2058471

6 53.4 0.0390824 0.2344945

7 53.4 0.0371012 0.2597085

8 53.4 0.0352204 0.2817635

9 53.4 0.0334350 0.3009151

10 1053.4 0.6261237 6.2612374

Total 8.0014280

81

m

iiwiD

Example: m = 10, c = 5.34%, y = 5.34%

Duration Hedging Example

• Duration of a zero coupon bond is– Equal to maturity

• For a given maturity and yield, duration increases as coupon rate– Decreases

• For a given coupon rate and yield, duration increases as maturity– Increases

• For a given maturity and coupon rate, duration increases as yield rate– Decreases

Duration Hedging Properties of Duration

Duration Hedging Properties of Duration - Example

Bond Maturity Coupon YTM Price Sens DBond 1 1 7% 6% 100.94 -0.94 1Bond 2 1 6% 6% 100 -0.94 1Bond 3 5 7% 6% 104.21 -4.15 4.40Bond 4 5 6% 6% 100 -4.21 4.47Bond 5 10 4% 6% 85.28 -7.81 8.28Bond 6 10 8% 6% 114.72 -7.02 7.45Bond 7 20 4% 6% 77.06 -12.47 13.22Bond 8 20 8% 7% 110.59 -10.32 11.05Bond 9 50 6% 6% 100 -15.76 16.71

Bond 10 50 0% 6% 5.43 -47.17 50.00

Duration Hedging Properties of Duration - Linearity

• Duration of a portfolio of n bonds

where wi is the weight of bond i in the portfolio, and:

• This is true if and only if all bonds have same yield, i.e., if yield curve is flat• If that is the case, in order to attain a given duration we only need two bonds

n

1iiiP wDD

1wn

1ii

• Principle: immunize the value of a bond portfolio with respect to changes in yield– Denote by P the value of the portfolio– Denote by H the value of the hedging instrument

• Hedging instrument may be – Bond – Swap – Future – Option

• Assume a flat yield curve

Duration Hedging Hedging

• Changes in value– Portfolio

Duration Hedging Hedging

0)(')(' dyyPyqHqdHdP

H

P

H

P

DurHDurP

SensHSensP

yHyPq

)(')('

dyyPdP )('

dyyHdH )('– Hedging instrument

• Strategy: hold q units of the hedging instrument so that

• Solution

• Example: – At date t, a portfolio P has a price $328635, a 5.143% yield and a

7.108 duration– Hedging instrument, a bond, has a price $118.786, a 4.779% yield

and a 5.748 duration

• Hedging strategy involves a buying/selling a number of bonds

q = -(328635x7.108)/(118.786x5.748) = - 3421

• If you hold the portfolio P, you want to sell 3421 units of bonds

Duration Hedging Hedging

• Duration hedging is– Very simple– Built on very restrictive assumptions

• Assumption 1: small changes in yield– The value of the portfolio could be approximated by its first order Taylor

expansion– OK when changes in yield are small, not OK otherwise– This is why the hedge portfolio should be re-adjusted reasonably often

• Assumption 2: the yield curve is flat at the origin – In particular we suppose that all bonds have the same yield rate– In other words, the interest rate risk is simply considered as a risk on the

general level of interest rates

• Assumption 3: the yield curve is flat at each point in time– In other words, we have assumed that the yield curve is only affected only

by a parallel shift

Duration Hedging Limits