Modelling Credit Spread Behaviour

FIRST

FIRST Credit, Insurance and RiskAngelo Arvanitis, Jon Gregory, Jean-Paul Laurent

ICBI Credit, Counterparty & Default Risk Forum29 September 1999, Paris

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OverviewOverview

Part I−Need for Credit Models

Part II−Simple Binomial Model

Part III−Jump-Diffusion Modelpage 2

Part IV−Credit Migration Model

Part V−Estimating Credit Spread Volatilities

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Part INeed for credit spread modelspage 3

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Need For Credit Models (I)Need For Credit Models (I)

- Credit derivatives market- Active management of loan portfolios

Why? Growth of emerging markets

page 4

Active management of counterparty risk in standard

derivatives portfolios

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Need For Credit Models (II)Need For Credit Models (II)

Valuing credit derivatives, options on risky bonds, vulnerable derivatives

Assessing the credit risk of portfolios -

spread and event riskWhat for?

page 5

- Optimising portfolio risk / return profile- Relative value analysis

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Need For Credit Models (III)Need For Credit Models (III)

Model the evolution of the risk free rate and

the credit spread

Estimate the current risk freeand risky term structures

Calibrate to observedbond and option prices

How?

page 6

Maturity

Yiel

d

Risk-freeRisky

Credit spread

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Credit DataCredit Data

Limited / crude data available on creditMoody’s historical data (annual)

−Default probability

−Pairwise default correlation

−Credit migration

−Loss given defaultDefault correlation and recovery rate difficult to

estimateCredit crashes - high default correlation

0 20%≤ ≤q kl

0 25%≤ ≤pi

0 5%≤ ≤ρ ij

0 100%≤ ≤li

page 7

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Credit spread for an AA bondCredit spread for an AA bond

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

80

90

Jump

page 8

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Properties of Credit SpreadsProperties of Credit Spreads

Jump Component- Discrete change in default

probability- Credit migration

Continuous Component- Mean reverting

- Change in market price of risk - risk premia

mean reversiondowngrade

Time

more volatile

Cre

dit s

prea

d

page 9

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Modelling Credit SpreadModelling Credit Spread

r rrisky risk free= + ~λ

Credit Spread

ConstantSimple binomial model

(Part II)

Continuous and jump components

Jump-diffusion model (Part III)

Model underlyingcredit migration process

(Part IV)

page 10

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Part IISimple Binomial Model

page 11

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Simple Binomial Model (I)Simple Binomial Model (I)

- Constant risk free term structure

- Constant recovery rate

- Constant credit spread if no default

- Jump in credit spread if default occurs

page 12

- Derive risk neutral default probabilities from risky and

risk- free bond prices

Risk neutral default probabilities - Actual default probabilities

- Risk premia- Liquidity

- Uncertainty over recovery rate

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Simple Binomial Model (II)Simple Binomial Model (II)

( )[ ]v T p T q T q T( ) ( ) ~( , ) ~( , )= − +1 0 0 δ

probability of default

default

v T( )risky bond price

1

δ

1 0− ~ ( , )q T

~ ( , )q T0

promised amount

recovery rate

no default

risk free bond price

~( , )( ) / ( )( )

q Tv T p T

01

1=

−− δ

- price any product with payoff contingent on default event

page 13

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Part IIIJump-Diffusion Model

page 14

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Jump-Diffusion ModelJump-Diffusion Model

Continuous component - Positive and mean reverting- Correlated with interest rates

Jump Component- Jumps of random size occur at random times

- Jumps in only one direction

Risk-free interest rate-Continuous and mean

reverting

- Standard implementation and calibration

- Standard numerical pricing algorithms can be used

page 15

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Risk Free Term Structure (I)Risk Free Term Structure (I)

Assumptions on the future evolution of the instantaneous risk free rate

−Volatility (normal, lognormal, square root, … )−Drift / mean reversion

Long term meanRate of mean reversion

σ r r t( )

r t( )k tr ( )

page 16

( )r t r k t r t r t dt r t dW tr

t

r r

t

( ) ( ) ( ) ( ) ( ) ( ) ( )= + − +∫ ∫00 0

σ

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Risk Free Term Structure (II)Risk Free Term Structure (II)

σ r rk= =0 1 2. , σ r rk= =0 1 1 0. ,

σ r rk= =0 1 2. , σ r rk= =0 2 2. ,

page 17

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Credit Spread Term Structure (I)Credit Spread Term Structure (I)

~( ) ( ) ( )λ ρt r t x t= + ( )corr r t x t( ), ( ) = 0

determines correlationbetween and

~( )λ t r t( )- Uncorrelated with interest rates- Continuous and jump component

page 18

θ θe zz− >, 0

e n nn− = =λτ λτ τ( ) / ! , , , . . . tim e interval0 1 2

Random jump size z, exponentially distributed

Random number of jumps - follows Poisson process

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Credit Spread Term Structure (II)Credit Spread Term Structure (II)

( )x t x k s x s x s dsx

t

( ) ( ) ( ) ( ) ( )= + −∫00

+ ∫σ x x

t

s x s dW s( ) ( ) ( )0

+≤

∑Z ii i t

( ); ( )τ

Credit spread component uncorrelated with the risk free interest rate

page 19

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Credit Spread Term Structure (III)Credit Spread Term Structure (III)

more frequentand larger jumps

page 20

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Part IVCredit Migration Model

page 21

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Credit Migration ModelCredit Migration Model

- Model jointly assets in various credit classes

- Portfolio management and risk analysis

- Jumps modelled as changes in credit ratings and defaults

- Continuous part modelled as continually changing risk premia

page 22

- Flexible in terms of data requirements and

number of states

- Calibration - incorporate economic and historical

information

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Markov Chains - Generator Matrix (I)Markov Chains - Generator Matrix (I)

absorbing state (default)

constantover time

~

~ ~ . ~ ~~ ~ . ~ ~

. . . . .~ ~ . ~ ~

.

,

,

, , ,

Λ =

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

−

−

− − − −

λ λ λ λλ λ λ λ

λ λ λ λ

1 12 1 1 1

21 2 2 1 2

1 1 1 2 1 1

0 0 0 0

K K

K K

K K K K K

1 2 K-1 K

12

K-1K

Continuous time Markov chainDiscrete state space

page 23

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Markov Chains - Generator Matrix (II)Markov Chains - Generator Matrix (II)

, transition matrix over short period dtI dt+ ~Λ

, non-negative transition probabilities

, sum of all probabilities equals 1

A state i+1 is always more risky than state i

λij ≥ 0

~ ~λ λi ijij i

K

= −=≠

∑1

~ ~ , ,,λ λijj i

K

i jj k

i k k i≥

+≥

∑ ∑≤ ∀ ≠ +1 1

page 24

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Markov Chains - Transition MatrixMarkov Chains - Transition Matrix

Transition matrix for the period t to TExplicit computation

[ ]~( , ) exp ( )Q t T D T t= −−Σ Σ1

~Λ Σ Σ= −1D

~( , )

~ ( , ) . ~ ( , ) ~ ( , )~ ( , ) . ~ ( , ) ~ ( , )

. . . .~ ( , ) . ~ ( , ) ~ ( , )

.

,

,

, ,

Q t T

q t T q t T q t Tq t T q t T q t T

q t T q t T q t T

K K

K K

K K K K

=

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

−

−

− − −

1 1 1 1

21 2 1 2

1 1 1 1

0 0 1

page 25

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Model Structure (I)Model Structure (I)

States : uniquely determine default probabilityCredit ratings - can incorporate past credit

rating transitions - non-Markovian model

- Risk neutral generator matrix

- constant

~Λ

~Λ

page 26

jump in credit spread due to downgrade

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Model Structure (II)Model Structure (II)

Incorporate stochastic risk premia

continuous process for risk premia

~Λstochastic

jump in credit spread due to downgrade

~ ~ ( )Λ Λstochastic U t= ×

page 27

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Stochastic Generator Matrix Stochastic Generator Matrix

Stochastic generator matrix arises from randomly changing risk premia

~ ~ ( )Λ Λstochastic U t= ×

( )U t U a kU t dt U t dWt

t

t

( ) ( ) ( ) ( )= + − +∫ ∫00 0

σ

Closed form formulae for bond prices

Stochastic component

Mean reverting process

page 28

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Stochastic Risk PremiaStochastic Risk Premia

If eigenvectors are constant, can poseΛ Σ Σ( , ) ( )t T t= −1D

Possible evolution of eigenvaluesdX a b X dt dw t X tj j j j j j j

K= − + = =( ) , ( ) diag( ( ))σ D 1

Pricing equation is now modified to

q t T X s ds tiK ij jt

T

jKj

K

( , ) ( ) exp ( ) ( )= ⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

−

=∫∑ Σ Σ1

1

E Dpage 29

Expectation has closed (algebraic) form− depends on parameters a b σ and on D(t)

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Calibration (I)Calibration (I)

Prices of risky bonds for various credit

classes and maturities- Least squares estimation- Adjust historical generator matrix to fit market prices- Achieve fit closest to historical data

B Ti ( , )0

- Historical generator matrix (estimated from one year transition

matrix)- Credit spread historical time series

Λ

~

( , , , )ΛΛ

stochastica k σ

Price exoticstructures

Simulate Credit Spread

Simulate Credit Spread

page 30

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Calibration (II)Calibration (II)

priorgenerator matrix

( )min ( ) ( ; ~ )

~

~,Λ

ΛP F h v hji

ji

h

Ti

j

J

i

Kij ij

iji j

Ki

−⎛⎝⎜

⎞⎠⎟ +

−⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪=== =∑∑∑ ∑

1

2

11

2

1

λ λβ

market price of coupon bond for

class i

coupon at date h

Least squares fit to match directly observed coupon bond prices (any number)

confidence level

page 31

Obtain solution closest to the historical generator matrix - stable calibrationΛ

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Calibration - Emerging Markets (II)Calibration - Emerging Markets (II)

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

3 states

5 states

Actual

page 32

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Calibration - Corporate Market (II)Calibration - Corporate Market (II)

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

AAA

AA

A

BBB

BB

B

CCC

Cre

dit s

prea

d (b

p)

Maturity (years)

page 33

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Calibration - US Industrials (I)Calibration - US Industrials (I)

0

20

40

60

80

100

120

140

160

180

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

AAA

AA

A

Cre

dit S

prea

d (b

p’s)

Maturity (years)

page 34

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Calibration - US Industrials (II)Calibration - US Industrials (II)

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

BBB

BB

B

CCC

Cre

dit S

prea

d (b

p’s)

Maturity (years)

page 35

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Part VEstimating Credit Spread Volatility

page 36

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Credit spread volatilities estimatesCredit spread volatilities estimates

74 Bonds67 Investment grade (Baa and above) US Industrial bonds7 Speculative grade (Ba and below) Emerging Market bonds

74 Bonds67 Investment grade (Baa and above) US Industrial bonds7 Speculative grade (Ba and below) Emerging Market bonds

8 Model states1 Moody’s Aaa 5 Moody’s Ba1 - Ba32 Moody’s Aa1 - Aa3 6 Moody’s B1 - B33 Moody’s A1 - A3 7 Moody’s CCC4 Moody’s Baa1 - Baa3 8 Default

8 Model states1 Moody’s Aaa 5 Moody’s Ba1 - Ba32 Moody’s Aa1 - Aa3 6 Moody’s B1 - B33 Moody’s A1 - A3 7 Moody’s CCC4 Moody’s Baa1 - Baa3 8 Default

page 37

Historical generator matrix from Moody’s average 1y transition matrix 1920 - 1996Historical generator matrix from Moody’s average 1y transition matrix 1920 - 1996

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Estimated short term spreadsEstimated short term spreads

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

Credit ratings

AA ABBB BB

B CCCpage 38

Lower ratings have− higher spread− higher volatility

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Eigenvalues of generator matrixEigenvalues of generator matrix

0 50 100 150 200 250-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 50 100 150 200 250-0 05

0

0.05

0 50 100 150 200 250-0.05

0

0.05

0 50 100 150 200 250-0.05

0

0.05

0 50 100 150 200 250-0.05

0

0.05

PrincipalcomponentsEigenvalues

- first 3 principal components account

for 99% of the variance

page 39

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Advanced Modelling IssuesAdvanced Modelling Issues

Stochastic Recovery Rates- Recovery rates are random with

high variance- Exogenous- Endogenous - depend

on the severity of default

Credit Events Correlatedwith Interest Rates- Credit migration and defaults

depend on interest rates- Joint state variables for interest

rates and credit spreads- Incorporate business cycles

Second Generation Products- Basket options - credit spread,

default correlations- Multiple Currencies- Quantos

Non - Markovian Bankruptcy Process- Autocorrelated migration

process- Markovian in state space

augmented with laggedvalues

page 40

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