Chapter 4

Deriving the Zero-Coupon Yield Curve

FIXED-INCOME SECURITIES

Outline

• General Principle• Spot Rates• Recovering the Term Structure• Direct Methods• Interpolation• Indirect Methods• Splines• Term Structure of Credit Spreads

Last Time

• The current price of a bond (P0) paying cash-flows Ft is given by:

T

tt

t

rFP

10 )1(

• Now, do we expect to get the same rate when borrowing/lending for a year versus 10 years?

• Not necessarily

• Term structure of interest rates

General Principle

• General formula

T

tt

T

tt

t

t tBFRFP

11 ,00 ,0

)1(

– R(0,t) is the discount rate– B(0,t) is the discount factor (present value of $1 received at date t)– Discount factor more convenient: no need to specify frequency

• What exactly does that equation mean?– Q1: Where do we get the B(0,t) or R(0,t) from?– Q2: Do we use the equation to obtain bond prices or implied

discount factors/discount rates?– Q3: Can we deviate from this simple rule? Why?

Spot Rates

• Q1: Where do we get the B(0,t) or R(0,t) from? – Any relevant information concerning how to price a security should be obtained from

market sources– More specifically, B(t,T) is the price at date t of a unit pure discount bond paying $1 at

date T

• Discount factor B(0,t) is the price of a T-Bond with unit face value and maturity t

• Spot rate R0,t is the annualized rate on a pure discount bond:

tBR t

t

,0)1(

1

,0

• Bad news is no such abundance of zero-coupon bonds exists in the real world

• Good news is we might still be able to compute the spot rate

Bond Pricing

• Answer to the “chicken-and-egg'' second question (Q2) is

– It depends on the situation– Roughly speaking, one would like to use the price of primitive securities as

given, and derive implied discount factors or discount rates from them– Then, one may use that information (more specifically the term structure of

discount rates) to price any other security– This is known as relative pricing

• Answer to the third question (Q3) is– Any deviation from the pricing rules would imply arbitrage opportunities– Practical illustrations of that concept shall be presented in what follows– Everything we cover in this Chapter can be regarded as some form of

perspective on these issues

Spot Rates

• Example of spot rate:– Consider a two-year pure discount bond that trades at $92– The two-year spot rate R0,2 is:

• The collection of all spot rates for all maturities is:– The Term Structure of Interest Rates

%26.4 ;)1(

10092 2,022,0

RR

Recovering the Term StructureDirect Methods - Principle

• Consider two securities (nominal $100):– One year pure discount bond selling at $95– Two year 8% bond selling at $99

• One-year spot rate:

• Two-year spot rate:

%26.5 ;)1(

10095 1,01,0

RR

%7.8 ;)1(

1080526.18

)1(108

)1(899 2,02

2,02

2,01,0

RRRR

Recovering the Term Structure Direct Methods - Principle

• We may “construct” a two year pure discount bond• Two components:

– Buy the two year bond– Shortsell .08 units of the one year bond

• Cost:4919508099 .).(

Recovering the Term Structure Direct Methods - Principle

• Schedule of payments:

Today 1 year 2 years

-91.4 0 108

• This is like a two-year pure discount bond• Two-year rate is again:

%7.8 ;)1(

1084.91 2,022,0

RR

Recovering the Term StructureDirect Methods - Example

• If you can find different bonds with same anniversary date, then you can directly get the spot rates :

Coupon Maturity (year) PriceBond 1 5 1 101Bond 2 5.5 2 101.5Bond 3 5 3 99Bond 4 6 4 100

• Solve the following system101 = 105 B(0,1)101.5 = 5.5 B(0,1) + 105.5 B(0,2) 99 = 5 B(0,1) + 5 B(0,2) + 105 B(0,3)100 = 6 B(0,1) + 6 B(0,2) + 6 B(0,3) + 106 B(0,4)

• And obtain B(0,1)=0.9619, B(0,2)=0.9114, B(0,3)=0.85363, B(0,4)= 0.7890R(0,1)=3.96%, R(0,2)=4.717%, R(0,3)=5.417%, R(0,4)=6.103%

• 1 year and 2 months ratex=5.41%• 1 year and 9 months ratey= 5.69%• 2 year ratez= 5.69%

Recovering the Term StructureBootstrap: Practical Way of Implementing Direct Method

Maturity ZCOvernight 4.40%1 month 4.50%2 months 4.60%3 months 4.70%6 months 4.90%9 months 5.00%1 year 5.10%

Coupon Maturity (years) PriceBond 1 5% 1 y and 2 m 103.7Bond 2 6% 1 y and 9 m 102Bond 3 5.50% 2 y 99.5

6/116/1 )1(105

%)6.41(57.103

x

12/9112/9 )1(6

%)51(6102

y

21 )1(5.105

%)1.51(5.55.99

z

Recovering the Term StructureInterpolation - Linear

• Interpolation– Term structure is a mapping -> R(t, ) for all possible – Need to interpolate

• Linear interpolation– We know discount rates for maturities t1 et t2

– We are looking for the rate with maturity t such that t1< t <t2

)(),0()(),0()(),0(

12

2112

tttRtttRtttR

%875.51

%675.0%5.525.0)75.3,0(

xR

•

• Example: R(0,3) =5.5% and R(0,4)=6%

Recovering the Term StructureInterpolation – Piecewise Polynomial

• Cubic interpolation for different segments of the term structure

– Define the first segment: maturities ranging from t1 to t4 (say 1 to 2 years)

– We know R(0, t1), R(0, t2), R(0, t3), R(0, t4)

• The discount rate R(0, t) is defined bydctbtattR 23),0(

dctbtattRdctbtattRdctbtattRdctbtattR

42

43

44

32

33

33

22

23

22

12

13

11

),0(),0(),0(),0(

• Impose the constraint that R(0, t1), R(0, t2), R(0, t3), R(0, t4) are on the curve

Recovering the Term StructurePiecewise Polynomial - Example

• We have computed the following rates– R(0,1) = 3%– R(0,2) = 5%– R(0,3) = 5.5%– R(0,4) = 6%

• Compute the 2.5 year rateR(0,2.5) = a x 2.53 + b x 2.52 + c x 2.51 + d = 5.34375%

with

02.007.00225.0

0025.0

%6%5.5

%5%3

1416641392712481111 1

dcba

Recovering the Term StructurePiecewise Polynomial versus Piecewise Linear

3.00%

3.50%

4.00%

4.50%

5.00%

5.50%

6.00%

6.50%

1 1.5 2 2.5 3 3.5 4Maturity

Rat

e

LinearCubic

• Rather than obtain a few points by boostrapping techniques, and then extrapolate, it usually is more robust to use a model for the yield curve

• So-called indirect methods involve the following steps – Step 1: select a set of K bonds with prices Pj paying cash-flows Fj(ti) at dates ti>t– Step 2: select a model for the functional form of the discount factors B(t,ti;ß), or the

discount rates R(t,ti;ß), where ß is a vector of unknown parameters, and generate prices

Recovering the Term StructureIndirect Methods

N

i

ttRtti

jN

iii

jj ijetFttBtFtP1

);,(

1

)();,()()(ˆ

2K

1j

jj tPtP

)()(ˆminarg

– Step 3: estimate the parameters ß as the ones making the theoretical prices as close as possible to market prices

• Nelson and Siegel have introduced a popular model for pure discount rates

Recovering the Term StructureIndirect Methods – Nelson Siegel

)exp()exp(1)exp(1,0( 210 τθτθ

τθβτθ

τθββθR

R(0,) : pure discount rate with maturity

0 : level parameter - the long-term rate

1 : slope parameter – the spread sort/long-term

2 : curvature parameter

1 : scale parameter

Recovering the Term StructureInspection of Nelson Siegel Functional

Inspection of Nelson-Siegel Functional Form

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

0 5 10 15 20 25

Long-term limit

(1-exp(-/))./: short-term component

Term structure

((1-exp(-/))./-exp(-/)) medium-term component

Recovering the Term StructureSlope and Curvature Parameters

• To investigate the influence of slope and curvature parameters in Nelson and Siegel, we perform the following experiment

– Start with a set of base case parameter values 0 = 7% 1 = -2% 2 = 1% = 3.33

– Then adjust the slope and curvature parameters 1 = between –6% and 6% 2 = between –6% and 6%

Recovering the Term StructureInitial Curve

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0 5 10 15 20 25 30

Maturity

Zero

-Cou

pon

Rate

Recovering the Term StructureImpact of Changes in the Slope Parameter

1.00%

3.00%

5.00%

7.00%

9.00%

11.00%

13.00%

0 5 10 15 20 25 30

Maturity

Zero

-Cou

pon

Rate

Recovering the Term StructureImpact of Changes in the Curvature Parameter

4.00%

4.50%

5.00%

5.50%

6.00%

6.50%

7.00%

7.50%

8.00%

8.50%

0 5 10 15 20 25 30

Maturity

Zero

-Cou

pon

Rat

e

Recovering the Term StructurePossible Shapes for the Yield Curve

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

Zer

o-co

upon

rat

e

Recovering the Term StructureEvolution of Parameters of the Nelson and Siegel Model on the

French Market - 1999-2000

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

04/01/1999 17/04/1999 29/07/1999 09/11/1999 20/02/2000 02/06/2000 13/09/2000 25/12/2000

Valu

e of

the

Para

met

ers

béta 0béta 1béta 2

Beta(0) oscillates between 5% and 7% and may be regarded as the very long term rateBeta(1) is the short to long term spread. It varies between -2% and -4% in 1999, and then decreases in absolute value to almost 0% at the end of 2000 Beta(2), the curvature parameter, is the more volatile parameter which varies from -5% to 0.7%.

• An augmented form exists

Recovering the Term StructureIndirect Methods – Augmented Nelson Siegel

R(0,) : pure discount rate with maturity

3 : level parameter

2 : scale parameter

Allows for more flexibility in the short end of the curve

)exp()exp(1

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

22

23

210

τθτθ

τθβ

τθτθ

τθβτθ

τθββθR

Recovering the Term StructureAugmented Nelson Siegel - Illustration

Augmented Nelson-Siegel

.035

.037

.039

.041

.043

.045

.047

.049

0 5 10 15 20 25

• These models are heavily used in practice• One key advantage is they are parsimonious

– Do not involve many parameters– This induces robustness and stability– Very important in the context of hedging

• One drawback is their lack of flexibility– Can not account for all possible shapes of the TS we see in practice

• Alternative approach: spline models– More flexible– Better for pricing– Less parsimonious

• Spline models come in different shapes– Cubic splines– Exponential splines– B-splines

Recovering the Term StructureParsimonious Models – Pros and Cons

• Discount factors as polynomial splines

Polynomial Splines

20,10,)(

10,5,)(5,0,)(

)(3

22

22210

31

21115

30

20000

ssasbscdsBssasbscdsBssasbscdsB

sB

2,1,01)0(

)10()10(

)5()5(

0

)(5

)(10

)(5

)(0

iB

BB

BBii

ii• Impose smooth-pasting constraints

• Cut down the number of parameters from 12 to 5

• Discount factors as B-splines

• Discount factors as exponential splines (vasicek and Fong (1982))

B-Splines and Exponential Splines

43

41

3

1

3

3

)(1),(

)(),(

p

pjj

p

jipi ji

n

pp

n

ppp

ttB

BttB

3

22

2222

31

21111

30

20000

)()()(

),(eaebecdBeaebecdBeaebecdB

ttB

Example (French Market)

Bond Maturity Coupon Market price Model Price Residuals % errorBTAN 12/03/2001 5.75 103.113 103.095 0.018 0.02%BTAN 12/07/2001 3 98.724 98.729 -0.005 0.00%BTAN 12/10/2001 5.5 105.332 105.310 0.022 0.02%BTAN 12/01/2002 4 101.106 101.102 0.005 0.00%BTAN 12/03/2002 4.75 101.710 101.706 0.004 0.00%BTAN 12/07/2002 4.5 99.526 99.540 -0.015 -0.01%OAT 25/11/2002 8.5 113.454 113.440 0.013 0.01%

BTAN 12/01/2003 5 102.843 102.874 -0.031 -0.03%OAT 25/04/2003 8.5 111.054 111.105 -0.051 -0.05%OAT 25/10/2003 6.75 110.295 110.314 -0.019 -0.02%

BTAN 07/12/2003 4.5 101.603 101.405 0.198 0.20%OAT 27/02/2004 8.25 113.785 113.858 -0.074 -0.06%

BTAN 12/07/2004 3.5 94.778 94.812 -0.035 -0.04%OAT 25/10/2004 6.75 111.515 111.532 -0.017 -0.02%OAT 25/04/2005 7.5 111.952 112.003 -0.051 -0.05%

BTAN 12/07/2005 5 99.910 99.918 -0.009 -0.01%OAT 25/10/2005 7.75 117.958 117.900 0.058 0.05%OAT 25/04/2006 7.25 112.312 112.309 0.003 0.00%OAT 25/10/2006 6.5 112.170 112.085 0.084 0.08%OAT 25/04/2007 5.5 103.239 103.312 -0.073 -0.07%OAT 25/10/2007 5.5 106.037 106.032 0.004 0.00%OAT 25/04/2008 5.25 101.559 101.574 -0.015 -0.02%OAT 25/10/2008 8.5 127.936 127.942 -0.006 0.00%OAT 25/04/2009 4 92.297 92.326 -0.029 -0.03%OAT 25/10/2009 4 93.832 93.817 0.015 0.02%OAT 25/04/2010 5.5 103.049 102.983 0.066 0.06%OAT 25/04/2011 6.5 111.331 111.380 -0.049 -0.04%

4.70%

4.80%

4.90%

5.00%

5.10%

5.20%

5.30%

0 2 4 6 8 10 12 14

Confidence interval

Yield curve on 09/01/00

Example

Term Structure of Credit Spreads Competing Methods

• The term structure of credit spreads for a given rating class and a given economic sector is needed to analyze the relative pricing of risky bonds

• It can be derived from market data through two different methods– Disjoint method: separately deriving the term structure of non-default zero-coupon

yields and the term structure of risky zero-coupon yields so as to obtain by differentiation the term structure of zero-coupon credit spreads

– Joint method: generating both term structures of zero-coupon yields through a one-step procedure

• There are two ways of modeling the relationship between discount factors in the context of joint methods– Additive spread:

– Multiplicative spread:

– Note that and ),(),(),( 0 stSstBstB ii

),(),(),( 0 stTstBstB ii 0),(0 stS 1),(0 stT

Term Structure of Credit Spreads Comparison - Example

• We derive the zero coupon spread curve for the bank sector in the Eurozone as of May 31th 2000, using the interbank zero-coupon curve as benchmark curve

• For that purpose, we use two different methods– Disjoint method: we consider the standard cubic B-splines to model the two

discount functions associated respectively to the risky zero-coupon yield curve and the benchmark curve; we consider the following splines [0;1], [1;5], [5;10] for the benchmark curve and [0,3], [3,10] for the risky class

– Joint method: we consider the joint method using an additive spread and use again the standard cubic B-splines to model the discount function associated to the benchmark curve and the spread function associated to the risky spread curve

Term Structure of Credit SpreadsComparison – Smoother Fit with Joint Method

Euro Bank Sector A-Swap 0 Coupon Spread

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7 8 9

Maturity

Spre

ad (i

n bp

s)

Disjoint EstimationMethod

Joint Estimation Method

• Credit spreads estimated through disjoint method may be unsmooth functions of time to maturity• This is consistent neither with common sense, nor with theoretical predictions (Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995), among others - see chapter 13) • On the other hand, quality of fit is usually higher with disjoint method: usual trade-off between quality-of-fit and robustness