the oxford guide to financial modeling by ho & lee chapter 3. bond market: the bond model the...

The Oxford Guide to Financial Modeling by Ho & Lee

Chapter 3. Bond Market:The Bond Model

The Oxford Guide to

Financial Modeling

Thomas S. Y. Ho and Sang Bin Lee

Copyright © 2004 by Thomas Ho and Sang Bin Lee. All rights reserved.

Chapter 3. Bond Markets: The Bond Model 2


3.1 Bond Mathematics

• Principal and coupons– Maturity– Coupons as a % of the principal– Perpetual bonds

• Accrued Interest– Quoted price and invoice price– Accruing linearly



3.1 Bond Mathematics (2)• Yield

– Yield to maturity

– Compounding yield: annual, quarterly, monthly… continuously

2(1 ) (1 ) (1 )T

coupon coupon coupon principalInvoice Price

YTM YTM YTM

2 2

2 2 2

(1 2) (1 2) (1 2) T

coupon coupon coupon principalInvoice Price

YTM YTM YTM

expPrice rT



3.2 Bonds and Bond Market

• Money Market– LIBOR rates

– Fed Fund rates

– Overnight Repo rates

LIBOR(month) 1 3 6 12

% 1.84 1.90 2.02 2.42

Discount rate 1.24

Fed Funds 1.50

Repo 1.68

Banker's acceptance

1.86

Prime rate 4.75



3.2 Bonds and Bond Market (2)• Treasure Securities

– Bill, notes, and bonds– STRIPS– TIPS

• Other Bonds– Corporates, Municipals, Mortgages…



3.3 Swap Market

• Counter parties in an exchange of payments, over the tenor of the swap

• Notional amount

• Vanilla swap: floating rate for the fixed rate

• The swap rate: the fixed rate for each swap tenor



Yield Curves

• Time value of money depending on the time of the payments: risk free rates

• Discount function: the present value factor• Nominal yield curve• Spot yield curve• Par yield curve



Spot Yield CurveFigure 3.1 Treasury market spot curve



3.4 Economics of the Yield Curve

• Real Rate and Nominal Rate– The Fisher Equation

• Yield Curve Shapes

nominal interest rate = real rate + expected inflation rate

Yield

Time to Maturity

Yield

Time to Maturity



3.4 Economics of the Yield Curve (2)

• Expectation Hypothesis– The expected interest rate = the forward rate

2

0,1 1,2 0,21 1 1r E r r

21,21.06 1 1.07E r

1,2 8.01%E r



3.4 Economics of the Yield Curve (3)

• Liquidity Premium Hypothesis– Upward sloping yield curve as explained by

the premium

• Preferred Habitat Hypothesis– Market structure affects the shape and

movement of the yield curve



3.5 Bond Model

• The bond cash flow is the combination of the coupon payments and the principal

• The cash flow is viewed as a portfolio of single payments

• The portfolio is the sum of the present value of each payment



3.5 Bond Model (2)

No ArbitrageOpportunity

Law of One Price



3.5 Bond Model (3)The cash flow for each year is given by: Term 1 2 3

Coupon 10 10 10

Principal

100

Cashflow

10 10 110

The price, by the law of one price, is

1( ) ( )T

iB P i CF i

( ) 10 0.9 10 0.8 110 0.7

94

Price P



3.5 Bond Model (4)10.5

...

... ...

... ...

...5 10 8010

2

c

2

c

2

c

2

c

2

cF

0 5 10 15 20 25 30Time to Maturity

0.2

0.4

0.6

0.8

1

tnuocsiDetaR

P 1 0.943

P 5 0.744

P 10 0.554



3.6 Forward Prices and Forward Rates

• Futures and Forward Contracts

– The marking to market mechanism of the futures market

– Forward delivery of a bond

– Arbitrage condition and the pricing of a forward contract

• Forward rate movement

– Forward rate under different shapes of the yield curve



Forward Pricing Model

• T* is the delivery date of the forward contract• T is the maturity of the zero coupon bond to be

delivered• P(·) is the discount function• F is the forward contract price base on $1 principal

( * ) ( *) ( *, )P T T P T F T T



Forward Pricing Model (2)

Time 0 T* T*+T

Holding a T*-year bond

P(T*)

Holding a T-year forward contract with a delivery date at year T*

Cash Flow P(T*) 0

*

0, *( *) 1T

T

coupon principal

P T r

( )coupon principal *, *( ) 1T

T T Tcoupon principal r

*

0, * *, ** 1 1T T

T T T TP T r r




* *

0 , * 0 , * *, *

1 1 1(1 ) (1 ) (1 )T T T T

T T T T T Tr r r

* 1 *

0 , * 1 0 , * *, * 1

1 1 1

(1 ) (1 ) (1 )T T

T T T Tr r r

** 10, * 1 0, * 1

*, * 1 0, * 1*0, * 0, *

(1 ) 11 (1 )

(1 ) 1

TTT T

T T TTT T

r rr r

r r

0, * 1 0, * *, * 1 0, * 1

0, * 1 0, * *, * 1 0, * 1

T T T T T

T T T T T

r r r r

r r r r




Timeto Maturity

tnuocsiDetaR

Spot

Forward

Timeto Maturity

tnuocsiDetaR

Spot

ForwardTimeto Maturity

tnuocsiDetaR

Spot

Forward



Forward Pricing Model (5)Maturity(T) 1 2 3

Spot rates(%) =8% =9% =10%

forward rates =10.01% =11.01% N/A

forward rates =12.03% N/A N/A

2 20,2

1,2

0,1

3 30,3

1,3

0,1

3 30,3

2,3 2 2

0,2

1 1.091 1 10.01%

1.081

1 1.11 1 11.01%

1.081

1 1.11 1 12.03%

1.091

rr

r

rr

r

rr

r



Forward Pricing Model (6)• Forward rate movements

( * )( *, )

( *)

P T TF T T

P T

( * )( *, ) ,

( *)

P T TF T T

P T

* ( )

( ) ,( )

P t TP T

P t

*

*

( * )( , *, )

( * )

P T T tF t T T

P T t

*

*

( * )( , *, )

( * )

( * ) ( * )

( ) ( )

( * )

( *)

P T T tF t T T

P T t

P t T T t P t T t

P t P t

P T T

P T

( * )( *, )

( *)

P T TF T T

P T



3.7 Bond Analysis

• Cheap/Rich Analysis– The valuation model determines the fair value of a bond.

– Cheap/rich = the observed price – the fair price

• Spot Yield Curve, Par Yield Curve, and Nominal Yield Curve– The spot curve determines the par curve

– The par curve determines the spot curve

– The discount function determines the spot and par curves

– Nominal yield curve derived from the observed prices



STRIPS

US STRIPS

Maturity Type Bid Price

Aug 02 ci 99 24/32

Aug 02 np 99 23/32

Nov 02 ci 99 14/32

Nov 11 ci 61 02/32



A standard statistical curve fitting methodology

2 3( )P T a bT cT dT

2( , , , ) iF a b c d

2 3 31 1

3 32 2

( ) 1 max[( ) ,0]

max[( ) ,0] ... max[( ) ,0]n n

P t at bt ct a t t

a t t a t t

Cubic spline function



Modified , Effective, Key Rate Duration

• Modified duration is related to the weighted average life of a bond

• Effective duration is the price sensitivity of a bond to the yield curve shifts

• The 2 risk measures are the same if the yield curve is flat and the bonds have no embedded option (ie the bond is a cash flow.)

• Key rate duration is the price sensitivity of a bond for each key rate shift



(Effective) duration

P effective duration spot yield

P



Modified Duration

0.51 2

TDuration

r



Key Rate Durations

Linear decreases in the size of the shift

0.044

0.046

0.048

0.05

0.052

0.054

0.25 0.5 1 2 5 7 10 30

Time to Maturity

Spo

t rat

e

Yield Curve

Shifted Curve

( ) ( )P

KRD i r iP



Key Rate Duration Profile

• The risk of a bond is measures by a set of key rate duration numbers

• The sum of key rate durations = effective duration

• Key rate duration of a zero coupon bond equals the duration at the bond maturity

Key Rate Duration of Zero Coupon Bond

9.7087 9.7087

02468

1012

0.25 0.5 1 2 5 10 30 dur

0.25

0.5

1

2

5

10

30

dur



Convexity

• Convexity provides the 2nd order approximation to the price behavior of a bond to the shift of the yield curve

• Convexity of a bond can be simulated using a bond valuation model

( (1 1

2

P P PConvexity

P

2

0.5 0.5( )P Duration P r Convexity P r



Performance Profile

• Performance profile relates the bond price to a range of parallel shifts of the yield curve

• The profile depicts the behavior of the bond, which can be complex, as described in later chapters.

-0.04-0.02 0 0.02 0.04

30

40

50

60

Zero coupon bond profile

Zer

o C

oup

on B

ond

P

rice

Parallel Shift



3.8 Applications of the Bond Analytics

• Barbell trade to enhance returns when the yield curve shifts in a parallel fashion

• Replicating a Treasury portfolio– For indexation– For enhance indexation– For asset liability management



A Barbell Trade

Bond position

valueMaturity Duration Convexity

A $100 1 0.9708 0.7069

B $100 5 4.8543 12.9606

Total $200 3 2.9125 6.8337

Short-selling $200 2.999 2.9125 4.9486



Appendix A: Taylor Expansion

Remainder

2 21( )

2f x

1( )f x

( )f x

x x

( )f x

x



Appendix A (2)

1 2 2

3 3

0

1 1( ) ( ) ( ) ( )

1! 2!1 1

( ) ( )3! !

1( )

!

( ) ( )

i i

i i

i

i

f x f x f x f x

f x f xi

f xi

where f x is the ith derivative of f x with respect to x



Appendix B: The Derivation of Macaulay Duration and Convexity



Appendix C: Duration & Convexity in measuring price sensitivity


The Oxford Guide to Financial Modeling by Ho & LeeRestrictive Assumptionsof Yield curve movements

ㆍparallel shifts

ㆍinfinitestimal shift

ㆍinstantaneous shifts( i.e, instantaneous investment horizon )

non parallel shifts a specific movements( e.g., rotations or inversions ) of the yield curve

finite shifts convexity

implied forward yield curvenot instantaneous at the end of the investment horizon investment horizon

non parallel shifts key rate duration ( vector )( e.g., rotations or inversions )

finite shifts key rate convexity ( matrix )

Effective duration

stochasticprocess risk

generalization ofduration (scalar)

andconvexity (scalar)

Relaxation of therestrictive

assumptions

the oxford guide to financial modeling by ho & lee chapter 3. bond market: the bond model the...

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