7-1

Chapter 7

Non Flat Term Structure

7-2

Notation

R0,1 = the spot interest rate observed at time 0 (first subscript) and lasting one period of time.

R0,2 = the spot interest rate observed at time 0 (first subscript) and lasting two periods.

7-3

R0,1Time

Spot interest rates

10 2

R0,2

7-4

One-period Present Value

PV1 =1

1 + R0,1

0.9615 =

11.04 1

10

$1

S1 = 96.15 =

1001.04 100

S1 = One-period strip

7-5

Two-period Present Value

PV2 =1

(1 + R0,2)2

S2 = Two-period strip

0.8573 =

1(1.08)2 1

10

$1

S2 = 85.73 =

100(1.08)2 100

2

7-6

n-period Present Value

PVn =1

(1 + R0,n)n

0.4665 =

1(1.10)8 1

0

$1

S8 = 46.65 =

100(1.10)8 100

n

7-7

1/(1 + R0,1) or 1/(1 + R0,2)2? or

Which of the following can be true?a. LHS > RHSb. LHS = RHSc. LHS < RHSd. It depends.e. None of the above.

11 + R0,1

1(1 + R0,2)2

7-8

PV1 PV2 . . . PVn

1 at timereceived $1 of

luePresent va

2 at timereceived $1 of

luePresent va

n at timereceived $1 of

luePresent va …

7-9

One-period Strip

Price ofone-period

strip= S1 = [PV1][PAR]

S1 = = 100 = 96.15.1,0R1

PAR

04.11

7-10

Two-period Strip

Price oftwo-period

strip= S2 = [PV2][PAR]

S2 = = 100 = 85.73.22,0 )R1(

PAR

2)08.1(

1

7-11

Cash flows for treasury strip with price Sn

Points in time

0 1 … n

Cash flows –Sn 0 … 0 … +PAR

7-12

strip. a of parof dollar per Price

PARstrip period-n of Price

n timeat received$1 of Value

present ThePARSPV n

n

7-13

Forward Interest Rates

7-14

Timing of a Forward Contract

Points in time

0 Delivery dateSign Pay $ and

contract receive commodityTransaction and or

Set Deliver commodity and price receive $

7-15

Forward Contract Cash Flows for Lender (Bond Buyer)

Points in time

0 1 2Delivery date Maturity date

Cash flows 0 -F +PAR(Lend $F) (Receive PAR)

7-16

Time 1 Value of $1 Received at Time 2

F =2,0f1

1

0.8917 =

11.1215 1

10

$1

2f0,2

7-17

Time 2 Value of $1 Received at Time 3

3,0f11

10

$1

3f0,3

2

= Time 2 value of $1 received at time 33,0f1

1

7-18

310 2

R0,1 f0,2 f0,3

Time 0 value = )f1)(f1)(R1(1

3,02,01,0

Time 1 value = )f1)(f1(1

3,02,0 of $1 received at time 3

of $1 received at time 3

7-19

Link between Spot and Forward

Interest Rates

7-20

210

R0,1 f0,2

1 + R0,1

R0,2

$1

$1 (1 + R0,1)(1 + f0,2)

(1 + R0,2)2

$1 1.04

$1

(1.04)(1.1215) = 1.1664

(1.08)2 = 1.1664

7-21

(1 + R0,2)2 = (1 + R0,1)(1 + f0,2).

Geometric mean of R0,1 and f0,2 is R0,2.

In terms of present value,

.)f1)(R1(1

)R1(1

2,01,02

2,0

7-22

.2fRR

fR1R21fRfR1RR21

2,01,02,0

2,01,02,0

2,01,02,01,02

2,02,0

Arithmetic Mean

Approximation

7-23

Arithmetic Mean

Approximation210

R0,1 f0,2

.04 .12+

2R0,2

2(.08) = .16

= .16

7-24

Three-period Case(1 + R0,3)3 = (1 + R0,1)(1 + f0,2)(1 + f0,3)(1 + R0,3)3 = (1 + R0,2)2 (1 + f0,3).

n-period Case(1 + R0,n)n = (1 + R0,1)(1 + f0,2) . . . (1+f0,n)

= (1 + R0,n-1)n-1 (1 + f0,n).

7-25

.R1)R1(f1

1,0

22,0

2,0

.)R1()R1(f1 2

2,0

33,0

3,0

.)R1()R1(f1 1n

1n,0

nn,0

n,0

In Terms of Forward Rates

7-26

Since the prices of one- and two-period strips are

20,2

20,1

1 )R(1PARS R1

PARS

.1215.173.8515.96

04.1)08.1(f1

SS

)R1(1R11

R1)R1(f1

2

2,0

2

1

22,0

1,0

1,0

22,0

2,0

7-27

.1

)1()1(

1

1,0

3

22

2,0

33,0

3,0

n

nn SSf

SS

RR

f

Similarly,

7-28

Shape of the Term Structure

(1 + R0,2)2 = (1 + R0,1)(1 + f0,2).

7-29Maturity1 2

Flat R0,1 R0,2

Flat Term Structure

= f0,2

7-30Maturity1 2

Rising R0,1

R0,2

R0,1 < R0,2

R0,1

Rising Term Structure

7-31Maturity1 2

Rising R0,1

R0,2

R0,1 < R0,2

R0,1

Rising Term Structure

f0,2

< f0,2

7-32Maturity1 2

Declining R0,1

R0,2

Declining Term Structure

R0,1

R0,1 > R0,2

f0,2

> f0,2

7-33

Annuities

7-34

.)R1(

1R11PVA Annuityof

luePresent Va2

2,01,02

.8188.18573.09615.0)08.1(

104.11

AnnuityofluePresent Va

2

.)R1(

1R11PVA Annuityof

luePresent Van

n,01,0n

7-35

Prices of Coupon-Bearing Bonds

Cash flows for Coupon-bearing bond

Points in time

0 1 2 through n–1 n

Cash flows –Pn c … c … +PAR

7-36

.PVPARcR1PARcP 1

1,01

One-period Bond

.92.10104.11006P1

7-37

.PVPARPVAc)R1(

PARcR1cP

22

22,01,0

2

Two-period Bond

.65.96)08.1(

100604.16P 22

7-38

.strip of Priceslopec

PVPARPVAc)R1(

PARcR1cP

nn

nn,01,0

n

n-period Bond

7-39

Yield to Maturity and Spot Rates

7-40

In general, y is a polynomial average of R0,1 and R0,2.

22,01,0

22 )R1(PARc

R1c

y)(1PARc

y1cP

7-41

Examples of Yield to Maturity0 1 2

Calculator: N = 2, PV = -96.65, PMT = 6, FV = 100.

y6 = 7.8754

96.65

+= 04.16

2)08.1(106

96.65

+= y16 2)y1(

106

7-42

0 1 2

y3 = 7.9360

Which bond is better?

91.19

+= 04.13

2)08.1(103

91.19

+= y13 2)y1(

103

7-43

Two-period Par Bond

%.84.71/(1.08)/(1.04)1(

)/(1.08)1(1yPAR

)R/(11)R/(11))R/(11(1yPAR

2

2

2

20,20,1

20,2

2

7-44

n-period Par Bond

. Annuityperiod-n of luePresent Van] Timeat $1 of luePresent Va[1

PVAPV1yPAR

)R/(11)R/(11))R/(11(1yPAR

n

nn

nn0,0,1

nn0,

n

7-45

Finding the Term Structure of

Interest Rates

7-46

STRIPS: One-period Strip

Since S1 = 1,0R1

PAR

1 + R0,1 = = = 1.041S

PAR15.96

100

R0,1 = 0.04 = 4%.

7-47

STRIPS: Two-period Strip

Since S2 = 22,0 )R1(

PAR

(1 + R0,2)2 = = = 1.16652S

PAR73.85

100

R0,2 = 0.08 = 8%.

7-48

STRIPS: n-period Strip

Since Sn = nn,0 )R1(

PAR

(1 + R0,n)n =nS

PAR

R0,n = – 1.nS

PARn

7-49

Bootstrapping—Recursive Method1-Period Bond

2-Period Bond

%5.4R 045.1

10452.99

R1ParcP

0,1

1,0

%5R )R(1

104.50045.150.409.99

)R1(Parc

R1cP

0,220,2

22,01,0

7-50

Disadvantages of Bootstrapping

Maturities may be missing.Errors in earlier maturities compound

and create bigger errors for longer maturities

7-51

Two Bonds with Same Maturity But Different Coupons

]PV[Par]PVA[cP

)R1(Parc

R1cP

22

22,01,0

7-52

P

c

S2

4.50

5.00

c99.415

100.35

P

0 21

104.504.5099.415105.005.00100.35

7-53

96.0PV 96S

10091

100S

87.1

ParS

ParS

PVA

00.91)87.1)(50.4(415.99S

87.150.400.5415.9935.100PVA

11

1

212

2

2

7-54

DISADVANTAGE

There may only be a small number of maturities with bond pairs or triples.