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An Introduction to Yield-Curve Modeling

Jean-Paul Renne

Banque de France

ENSTA

Jean-Paul Renne (Banque de France) An Introduction to Yield-Curve Modeling ENSTA 1 / 58


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Overview of the presentation

1 Generalities about fixed-income securities and yields

1 What is a fixed-income security?2 The yields3 Duration of a bond

4 Modeling the yield curve: first statistical approaches1 Principal component analysis2 Nelson-Siegel model

2 What accounts for the shape of the yield curve?

1 A world without uncertainty2 A world with uncertainty but without risk-aversion3 A more realistic world...



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What is a fixed income security?



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What is a fixed income security?

Capital markets are markets for securities (debt or equity), where

entities (business enterprises or governments) can raise long-termfunds.

In these markets, money is provided for periods longer than a year(raising short-term funds takes place on other markets, e.g., themoney market).

The capital market includes thestock market(equity securities) andthebond market(debt).

Capital markets may be classified as primary marketsandsecondarymarkets.

In primary markets, new stock or bond issues are sold to investors via amechanism known as underwriting.In the secondary markets, existing securities are sold and boughtamong investors or traders, usually on a securities exchange,over-the-counter, or elsewhere.



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Source: Bloomberg.Jean-Paul Renne (Banque de France) An Introduction to Yield-Curve Modeling ENSTA 6 / 58


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What is a yield-to-maturity?

Lets denote with P(t, h) the price of a riskless h-year bond payingannually a coupon ofc(in percentage point) and with a principal of100 (that will be paid in h years).

Theyield-to-maturity Rht (for the maturity h) is defined as that yieldthat is such that

P(t, h) =

hi=1

c

(1+Rht)i

+

100

(1+Rht)h

ExerciseAssume that P(t, h) =100 (that is, the bond is at par). Then, whatis the relationship between c and Rht?

IfP(t, h)< 100 , what is the relationship between c and Rht?



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Duration of a bond

The duration Dis a weighted average of the maturities of the pay-offsassociated to the bond.

The modified duration DM is defined as D/(1+Rht). Therefore

P(t, h)

P(t, h) = DMR

ht

Exercise

What is the duration of a zero-coupon bond if maturity h?



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Bond price and yield

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Stylized facts about yields

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CNO TEC indices (French government yields), Source: Datastream.



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Stylized facts about yieldsCorrelation matrix of French yields (1999-2009)

1Mth Yd 3Mth Yd 6Mth Yd 1Y Yd 2Y Yd 5Y Yd 10Y Yd

1Mth Yield 13Mth Yield 0.996 16Mth Yield 0.984 0.993 1

1Y Yield 0.955 0.971 0.991 12Y Yield 0.907 0.928 0.958 0.982 15Y Yield 0.786 0.802 0.835 0.863 0.929 1

10Y Yield 0.613 0.618 0.640 0.654 0.746 0.931 1



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Zero-coupon bondsPayoffs schedule

Azero-coupon bond(also known as a discount bond ) makes a single

payment on its maturity date.Acoupon bondmakes periodic interest payments prior to its maturitywhen it also makes a final payment that represents repayment ofprincipal ( a portfolio of zero-coupon bonds).

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What accounts for the shape of the yield curve?


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The pricing kernel


What accounts for the shape of the yield curve?


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The absence of arbitrage

In recent years the theory of finance has produced convenient toolsthat allow us to directly apply the conditions that guarantee theabsence of arbitrage opportunitiesin a world where there isuncertainty.

The tools were developed as an outgrowth of the famous

BlackScholes model of option prices.An arbitrage involves trading securities in such a way as to generatesomething for nothing.

The most powerful tool for understanding the term structure of

interest rates is called the absence of arbitrage. This is short-handfor the conditions that guarantee the absence of arbitrageopportunities. An opportunity for arbitrage exists when there is aninconsistency in the prices of securities that allows a valuable payoff tobe obtained at no cost.



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Macro-finance modelsG l f k (2/ )


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General framework (2/4)

Ingredients:

The dynamics of factors (observable or not) FtThe specifications of thepricing kernel mt+1 (orstochastic discountfactor,SDF)

Price Ptof an asset providing the payoffg(FT) in period T:

Pt=Et(mt+1mt+2 . . . mT1mTg(FT))

In Affine Term Structure Models (ATSM), zero-coupon bond yields ofmaturity are given by:

i1,ti2,t

...in,t

= A+BFt



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Macro-finance modelsG l f k (4/4)


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General framework (4/4)

Example 2: Pricing of a 1-period inflation-linked bond

The framework makes it possible to price inflation-linked bonds (ILBs)as soon as inflation is one of the factors Ft.

Lets inflation t= ln(CPIt/CPIt1) be the first component ofFt,then t= Ft where = 1 0 . . . 0 .

Payoff: g(Ft+1) = CPIt+1

CPIt, therefore Pt=Et

mt+1

CPIt+1CPIt

i1,t r1,t= Ft expected

inflation

(=Et(t+1))

1

2

convexity

adjustment

(0+1Ft) riskpremium



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Figure: Model structure

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The model broadly follows the lines of Rudebusch and Wus (2008) model



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ModelSpecifications: Phillips and IS curves


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Specifications: Phillips and IS curves

Phillips curve:

t=Lt+(

t1 Lt1) +yyt1+,t

Investment-saving (IS) curve:

yt=y(L)yt1 r(i1,t1 Et1(t)) +y,tThese first equations form a VAR that reads:

Ft= Ft1+ t

The stochastic shocks tare assumed to be normally i.i.d.



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Figure: Estimation data


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6

Jan03Jan04Jan05 Jan06Jan07Jan08Jan09Jan99 Jan00Jan01Jan02-4

-2

0

2

4

8

10

y-o-y inflation

Inflation and SPF expected inflation

monthly inflation

Jan03Jan02Jan01Jan00 Jan07Jan06 Jan08 Jan09-8

-4

0

4

8

Jan99 Jan05Jan04

Real activity

Jan06 Jan07Jan08 Jan090

1

2

3

4

5

6

Jan00Jan99 Jan01Jan02Jan03Jan04 Jan05

Nominal yields

-2

-1

0

1

2

3

4

Jan99 Jan00 Jan01Jan02Jan03Jan04 Jan05Jan06 Jan07Jan08 Jan09

Real yields


ModelData: Macroeconomic data


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Data: Macroeconomic data

The data cover the period from January 1999 to June 2009 at themonthly frequency (Eurozone data)

Real activity is represented by the first principal component of a set of5 business and consumer confidence indicators (source: EuropeanCommisson qualitative survey): industrial, construction, retail trade,service and consumer confidence

The inflation series (HICP excl. tobacco, source: Eurostat) isseasonally adjusted using Census X12

Inflation forecasts of the ECB Survey of Professional Forecastersare

included amongst the estimation series (3 additional measurementequations: SPF forecasts= model-implied expectations+ error term,for 1-, 2- and 5-year horizons)


ModelData: Interest rates


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Zero-coupon nominal and real (end-of-month) interest rates arederived from

Government-bond yields (bootstrap on a spline-smoothened FrenchTEC yield curve) and

inflation swap quotes (source: Bloomberg)Real yields are obtained as the difference between nominal yields andinflation swap rates (corrected from lags inherent in Eurozone inflationswaps)

The maturities of the zero-coupon bonds are as follows:

Nominal: 1, 3 and 6 months, 1, 2, 3, 5, 7 and 10 yearsReal: 1, 2, 5 and 10 years


ModelEstimation: A 2-step estimation procedure


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p p

In thefirst step, the macro-model parameters (+ the SPF error-termstandard deviation) are estimated by maximizing the log-likelihood(the log-L is obtained by applying the Kalman filter)

Three parameters have been calibrated (the inflation parameter g

entering the Taylor rule, the two parameters defining the dynamics ofmedium-term inflation, L and )

In asecond step, the state-space model is enlarged by adding nominaland real yields amongst the observed variables (the state-space modelis enlarged; all yields are assumed to be measured with errors)

The coefficients of the market price of risk (1 matrix) that load onlagged macro variables are set to zero



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ModelEstimation: Parameters


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1 y 103

1 4 r y103

0.21 0.043 1.52 1.14 -0.14 0.06 0.35

(0.05) (0.013) (0.1) (0.04) (0.04) (0.03) (0.02)

S g gy S103

L L103

0.95 0.50 0.83 0.117 0.95 0.50 0.050

(0.018) (-) (0.24) (0.009) (-) (-) (0.01)

3mth104

6mth104

1yr104

2yr104

3yr104

5yr104

7yr104

10yr104

0.78 1.21 2.00 2.35 1.83 1.27 0.84 2.17

(0.05) (0.08) (0.13) (0.15) (0.12) (0.09) (0.08) (0.15)r1yr104

r2yr104

r5yr104

r10yr104

SPF1yr104

SPF2yr104

SPF5yr104

4.82 4.35 2.91 2.20 0.88 0.60 0.38

(0.44) (0.37) (0.26) (0.2) (0.13) (0.07) (0.05)


Figure: Yield fit


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Figure: Yield fit

Jan09Jan07Jan05Jan03Jan012

3

4

5

6

Jan99

5-yr zero-coupon yield (observed)

5-yr zero-coupon yield (simulated)

Jan092

3

4

5

6

7

Jan07Jan05Jan03Jan01Jan99

10-yr zero-coupon yield (simulate

10-yr zero-coupon yield (observed


Figure: Model properties: Risk premiums


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in

bp

-50

0

50

100

150

200

250

300

Jan99 Jan00 Jan01Jan02Jan03Jan04Jan05Jan06 Jan07Jan08Jan09

5Y risk premiums

Inflation risk premium

Term premium

Jan04

in

%

Jan09Jan08Jan07Jan06Jan05Jan03Jan02Jan01Jan00Jan99

4

3

2

1

0

5Y model-implied expected inflation and SPF

Model-implied expected infl.

0.25

0.75

0.5

1

0

48

in

bp

maturity (in months)

120108968472-200

-100

0

100

200

300

400

500

0 12 24 36 60

Unconditional risk premiums

-2

0

1

2

3

4

5

6

7

8

60 10872

in

%

84 12096

maturity (in months)

483624120

-1

Unconditional yields


References


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Duffie, D. and Kan, R. (1996). A yield factor model of interest rates.Mathematical Finance, vol. 6.

Fisher, M. (2001). Forces That Shape the Yield Curve. WorkingPaper Series of the Federal Reserve Bank of Atlanta, No 2001-3.

Renne, J.-P. (2009). A frequency-domain analysis of debt service in amacro-finance model for the euro area. Banque de France WorkingPaper Series, No 261.

Rudebusch, G. and Wu, T. (2008). A macro-finance model of theterm-structure, monetary policy and the economy. The EconomicJournal, vol. 118.


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