[1] Pricing and HedgingMotivation

• 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 lecture

– 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 payingoff $1 at date T (« discount factor »)

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

!

B(t, t + ") =1

(1+ Ra(t,"))"

!

R(t,") = #1

"ln B(t,t + ")( )

!

B(t, t + ") = exp #" $ R(t,")( )

• R(t,θ) : continuously compounded pure discount ratewith maturity t + θ:

– Equivalently,

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

• Example: $100 bond with a 5% coupon

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

!

V (t) = FiB(t,t + i) =

i=1

m

"Fi

1+ Ra(t,i)[ ]

i

i=1

m

"

Pricing and HedgingPricing Certain Cash-Flows

• Example– Assume today a flat structure of interest rates– Ra(0,θ) = 10% for all θ– 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 HedgingInterest Rate Risk

• Basic principle: attempt to reduce as much aspossible 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 HedgingHedging Principles

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

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

• Price of the bond: (actually y/2)

Duration HedgingDuration

!

V =Fi

1+ y( )i

i=1

m

"

Duration HedgingSensitivity

)()( yVdyyVdV !+=

dyyVdV )('!

!

dV

V"V '(y)

V (y)dy =Sens# dy

• 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 ratevalue dy

• For small changes, can be approximated by

• Relative variation

• The relative sensitivity, denoted as Sens, is thepartial derivative of the bond price with respect toyield, divided by the bond price

• Formally

Duration HedgingDuration

!

Sens =V '(y)

V (y)=

"1

1+ y

iFi

1+ y( )i

i=1

m

#

V (y)

• In plain English: tells you how much relative changein 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) isreferred 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 HedgingTerminology

• Definition of Duration D:

• Also known as “Macaulay duration”• It is a measure of average maturity• Relationship with sensitivity and modified duration:

Duration HedgingDuration

8

1

!"=#=

m

i

iwiD

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

Duration HedgingExample

Time of

Cash Flow (i)

Cash Flow

Fi ( )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

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

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

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

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

Duration HedgingProperties of Duration

Duration HedgingProperties 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 durationwe only need two bonds

• Principle: immunize the value of a bond portfolio withrespect 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 HedgingHedging

• Changes in value– Portfolio

Duration HedgingHedging

!

dP + qdH = qH '(y) + P'(y)( )dy = 0

!

q = "P'(y)

H '(y)="P #Sens

P

H #SensH

="P #Dur

P

H #DurH

!

dP " P'(y)dy

dyyHdH )('!– Hedging instrument

• Strategy: hold q units of the hedging instrument sothat

• 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 numberof bonds

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

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

Duration HedgingHedging

• 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 affected only by a

parallel shift

Duration HedgingLimits

[2] Accounting for Larger Changes in YieldDuration and Interest Rate Risk

• Relationship between price and yield is convex:

• Taylor approximation:

Accounting for Larger Changes in YieldConvexity

!

"V

V#V '(y)

V (y)"y +

1

2

V"(y)

V (y)"y( )

2=Sens$"y +

1

2Conv $ "y( )

2

• Relative change

• Conv is relative convexity, i.e., the second derivativeof value with respect to yield divided by value

• $ Convexity = V’’(y) = Conv x V(y)• Example (back to previous)

– 10 year maturity bond, with a 6% annual coupon rate, a 7.36modified duration, a 6974 $ convexity and which sells at par

– Case 2: yields go from 6% to 8%

• Second order approximation to change in price– Find: -14.72 + (6974.(0.02)²/2) = -$13.33– Exact solution is -$13.42 and first order approximation is -$14.72

Accounting for Larger Changes in YieldConvexity and $ Convexity

!

Conv =V" y( )V y( )

=

1

1+ y( )2

i(i +1)Fi

1+ y( )i

i=1

m

"

V y( )

• (Relative) convexity is

Accounting for Larger Changes in YieldProperties of Convexity - Linearity

• Convexity 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

Accounting for Larger Changes inYield

Duration-Convexity Hedging• Principle: immunize the value of a bond portfolio

with respect to changes in yield– Denote by P the value of the portfolio– Denote by H1 and H2 the value of two hedging instruments– Needs two hedging instrument because want to hedge one risk

factor (still assume a flat yield curve) up to the second order

• Changes in value– Portfolio

!

dP " P'(y)dy +P' '(y)

2dy

2

!

dH1" H

1'(y)dy +

1

2H1' '(y)dy

2

dH2" H

2'(y)dy +

1

2H2' '(y)dy

2

#

$ %

& %

– Hedging instruments

Accounting for Larger Changes in YieldDuration-Convexity Hedging

• Strategy: hold q1 (resp. q2) units of the first (resp.second) hedging instrument so that

02211=!+!+ dHqdHqdP

!"#

=++

=++

0)('')('')(''

0)(')(')('

2211

2211

yHqyHqyP

yHqyHqyP

– Or (under the assumption of a unique y – flat yield curve)

!

q1H1(y)Dur

1+ q

2H2(y)Dur

2= "P(y)Dur

P

q1H1(y)Conv

1+ q

2H2(y)Conv

2= "P(y)Conv

P

# $ %

• Solution (under the assumption of unique dy –parallel shifts)

Accounting for a Non Flat Yield CurveAllowing for a Term Structure

• Problem with the previous method: we haveassumed a unique yield for all instrument, i.e., wehave assumed a flat yield curve

• We now relax this simplifying assumption andconsider 3 potentially different yields y, y1, y2

• On the other hand, we maintain the assumption ofparallel shifts, i.e., we assume dy = dy1 = dy2

• We are still looking for q1 and q2 such that

02211=!+!+ dHqdHqdP

Accounting for a Non Flat Yield CurveAccounting for a Non Flat Yield Curve

• Solution (under the assumption of unique dy –parallel shifts)

!

P '(y) + q1H1'(y

1) + q

2H2'(y

2) = 0

P ' '(y) + q1H1' '(y

1) + q

2H2' '(y

2) = 0

" # $

– Or (relaxing the assumption of a flat yield curve)

!

q1H1(y1)Sens

1+ q

2H2(y

2)Sens

2= "P(y)Sens

P

q1H1(y1)Conv

1+ q

2H2(y

2)Conv

2= "P(y)Conv

P

# $ %

– Just replace (Macaulay) duration by sensitivity or modified durationin the first equation

Accounting for a Non Flat Yield CurveTime for an Example!

• Portfolio at date t– Price P = $ 32863.5– Yield y = 5.143%– Sens = 6.76– Conv = 85.329

• Hedging instrument 1– Price H1 = $ 97.962– Yield y1 = 5.232 %– Sens1 = 8.813– Conv1 = 99.081

• Hedging instrument 2:– Price H2 = $ 108.039– Yield y2 = 4.097%– Sens2 = 2.704– Conv2 = 10.168

Accounting for a Non Flat Yield CurveTime for an Example!

• Optimal quantities q1 and q2 of each hedginginstrument are given by

!"#

$%=$$+$$$%=$$+$$

329.855.32863039.108168.10962.97081.99

76.65.32863039.108704.2962.97813.8

21

21

qq

qq

– Or q1 = -305 and q2 = 140

• If you hold the portfolio, you should sell 305 units ofH1 and buy 140 units of H2

Accounting for Non Parallel ShiftsAccounting for Changes in Shape of the TS

• Bad news is: not only the yield curve is not flat, but also itchanges shape!

• Aforementioned methods do not allow to account for suchdeformations– Additional risk factors– One has to regroup different risk factors to reduce the dimensionality of

the problem: e.g., a short, medium and long maturity factors• Systematic approach: factor analysis on historical data has

shed some light on the dynamics of the yield curve• 3 factors account for more than 90% of the variations

– Level factor– Slope factor– Curvature factor

Accounting for Non Parallel ShiftsAccounting for Non Parallel Shits

• To properly account for the changes in the yieldcurve, one has to get back to pure discount rates

!

V (t) = F(i)B(t,t + i) =i=1

m

"F(i)

1+ Ra(t,i)[ ]

i

i=1

m

"

!

V (t) = F(i)B(t,t + i) =i=1

m

" F(i)exp #i $ R(t,i)[ ]i=1

m

"

• Or, using continuously compounded rates

Accounting for Non Parallel Shifts Nelson-Siegel Model

• The challenge is that we are now facing m risk factors• Reduce the dimensionality of the problem by writing discount

rates as a function of 3 parameters• One classic model is Nelson and Siegel’s

!"

#$%

&''

''+!

"

#$%

& ''+=) )exp(

)exp(1)exp(1,0( 210 ()

()

()*

()

()**)R

– with R(0,θ): pure discount rate with maturity θ– β0 : level factor– β1 : slope factor– β2 : curvature factor– τ : fixed scaling parameter

• Hedging principle: immunize the portfolio with respect tochanges in the value of the 3 parameters

Accounting for Non Parallel ShiftsNelson Siegel Model

• Mechanics of the model: changes in betaparameters imply changes in discount rates, whichin turn imply changes in prices

• One may easily compute the sensitivity (partialderivative) of R(0,θ) with respect to each parameterbeta (see next slide)

• Very consistent with factor analysis of interest ratesin the sense that they can be regarded as level,slope and curvature factors, respectively

Accounting for Non Parallel ShiftsNelson Siegel

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Maturity of rates

Sen

siti

vit

y o

f ra

tes

béta 0

béta 1

béta 2

Accounting for Non Parallel ShiftsNelson Siegel Model

• Let us consider at date t=0 a bond with price P delivering thefuture cash-flows Fi

• The price is given by

!

P0

= Fi

i

" B(0,#i) = F

i

i

" e$#

iR (0,#

i)

!!!!

"

!!!!

#

$

%&

'()

*++

+++=

,

,=

%&

'()

* +++=

,

,=

+=,

,=

+

+

+

-

-

-

),0(

2

02

),0(

1

01

),0(

0

00

)/exp(/

)/exp(1

/

)/exp(1

ii

ii

ii

R

i

ii

i

i

i

R

i

i

i

i

i

R

i

ii

eFP

S

eFP

S

eFP

S

..

..

..

/./.

/..

0

/.

/..

0

.0

• Sensitivities of the bond price with respect to each betaparameter are

Accounting for Non Parallel ShiftsExample

• At date t=0, parameters are estimated (fitted) to be

Beta 0 Beta 1 Beta 2 Scale parameter8% -3% -1% 3

Maturity Coupon Price S0 S1 S2

Bond 1 2 ans 5% 98.627$ -192.51 -141.08 -41.28

Bond 2 7 ans 5% 90.786$ -545.42 -224.78 -156.73

Bond 3 10 ans 5% 79.606$ -812.61 -207.2 -173.03

Portfolio -1550.54 -573.06 -371.04

• Sensitivities of 3 bonds with respect to each beta parameter,as well as that of the portfolio invested in the 3 bonds, are

Accounting for Non ParallelShifts

Hedging with Nelson Siegel• Principle: immunize the value of a bond portfolio

with respect to changes in parameters of the model– Denote by P the value of the portfolio– Denote by H1, H2 and H3 the value of three hedging instruments– Needs 3 hedging instruments because want to hedge 3 risk

factors (up to the first order)– Can also impose dollar neutrality constraint q0H0 + q1H1 + q2H2 +

q3H3 + q4H4 = - P (need a 4th instrument for that)

• Formally, look for q1, q2 and q3 such that

!!!

"

!!!

#

$

=%

%+

%

%+

%

%+

%

%

=%

%+

%

%+

%

%+

%

%

=%

%+

%

%+

%

%+

%

%

0

0

0

2

33

2

22

2

11

2

1

33

1

22

1

11

1

0

33

0

22

0

11

0

&&&&

&&&&

&&&&

Gq

Gq

Gq

P

Gq

Gq

Gq

P

Gq

Gq

Gq

P

Beyond DurationGeneral Comments

• Whatever the method used, duration, modified duration,convexity and sensitivity to Nelson and Siegel parameters aretime-varying quantities– Given that their value directly impact the quantities of hedging instruments,

hedging strategies are dynamic strategies– Re-balancement should occur to adjust the hedging portfolio so that it

reflects the current market conditions• In the context of Nelson and Siegel model, one may elect to

partially hedge the portfolio with respect to some betaparameters– This is a way to speculate on changes in some factors; it is known as

« semi-hedging » strategies– For example, a portfolio bond holder who anticipates a decrease in

interest rates may choose to hedge with respect to parameters beta 1 andbeta 2 (slope and curvature factors) while remaining voluntarily exposed toa change in the beta 0 parameter (level factor)