FIRST Modelling Credit Spread Behaviour
FIRST Credit, Insurance and Risk Angelo Arvanitis, Jon Gregory, Jean-Paul Laurent ICBI Credit, Counterparty & Default Risk Forum 29 September 1999, Paris
Overview Overview yPart I
−Need for Credit Models
yPart II
−Simple Binomial Model
yPart III page 2
−Jump-Diffusion Model
yPart IV
−Credit Migration Model
FIRST Credit, Insurance & Risk
yPart V
−Estimating Credit Spread Volatilities Credit Modelling
Part I page 3
Need for credit spread models
FIRST Credit, Insurance & Risk
Credit Modelling
Need Need For For Credit Credit Models Models (I) (I) - Credit derivatives market - Active management of loan portfolios
Growth of emerging markets
Why? page 4
Active management of counterparty risk in standard derivatives portfolios
FIRST Credit, Insurance & Risk
Credit Modelling
Need Need For For Credit Credit Models Models (II) (II) Valuing credit derivatives, options on risky bonds, vulnerable derivatives
Assessing the credit risk of portfolios spread and event risk
What for? page 5
- Optimising portfolio risk / return profile - Relative value analysis
FIRST Credit, Insurance & Risk
Credit Modelling
Need Need For For Credit Credit Models Models (III) (III) Estimate the current risk free and risky term structures Model the evolution of the risk free rate and the credit spread
How? Calibrate to observed bond and option prices Yield
page 6
FIRST Credit, Insurance & Risk
it spr Cred
ea d
Risky Risk-free Maturity
Credit Modelling
Credit Credit Data Data yLimited / crude data available on credit yMoody’s historical data (annual) −Default probability
0 ≤ p i ≤ 25%
−Pairwise default correlation 0 ≤ page 7
ρ ij ≤ 5%
−Credit migration
0 ≤ q kl ≤ 20%
−Loss given default
0 ≤ l i ≤ 100%
yDefault correlation and recovery rate difficult to FIRST Credit, Insurance & Risk
estimate yCredit crashes - high default correlation Credit Modelling
Credit Credit spread spread for for an an AA AA bond bond 90
80
Jump
70
60
50
40
page 8
30
20
10
0
0
50
100
150
200
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Credit Modelling
250
300
Properties Properties of of Credit Credit Spreads Spreads
Credit spread
more volatile
Jump Component - Discrete change in default probability - Credit migration
mean reversion downgrade
page 9
Time
FIRST Credit, Insurance & Risk
Continuous Component - Mean reverting - Change in market price of risk - risk premia
Credit Modelling
Modelling Modelling Credit Credit Spread Spread ~
rrisky = rrisk free + λ Credit Spread
page 10
Continuous and jump components Jump-diffusion model (Part III)
Constant Simple binomial model (Part II)
FIRST Credit, Insurance & Risk
Model underlying credit migration process (Part IV) Credit Modelling
page 11
Part II Simple Binomial Model
FIRST Credit, Insurance & Risk
Credit Modelling
Simple Simple Binomial Binomial Model Model (I) (I) - Constant credit spread if no default - Jump in credit spread if default occurs
- Constant risk free term structure - Constant recovery rate
page 12
FIRST Credit, Insurance & Risk
- 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
Credit Modelling
Simple Simple Binomial Binomial Model Model (II) (II) 1 promised amount
no default
δ recovery rate
default
1 − q~ ( 0 , T )
risky bond price
v (T )
page 13
risk free bond price
q~ ( 0 , T ) probability of default
[
v ( T ) = p ( T ) (1 − q~ ( 0 , T ) ) + q~ ( 0 , T )δ
FIRST Credit, Insurance & Risk
]
1 − v (T ) / p(T ) ~ q ( 0, T ) = (1 − δ ) - price any product with payoff contingent on default event Credit Modelling
page 14
Part III Jump-Diffusion Model
FIRST Credit, Insurance & Risk
Credit Modelling
Jump-Diffusion Jump-Diffusion Model 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
page 15
FIRST Credit, Insurance & Risk
- Standard implementation and calibration - Standard numerical pricing algorithms can be used
Risk-free interest rate -Continuous and mean reverting
Credit Modelling
Risk Risk Free Free Term Term Structure Structure (I) (I) yAssumptions on the future evolution of the
page 16
instantaneous risk free rate −Volatility σ r r ( t ) (normal, lognormal, square root, … ) −Drift / mean reversion yLong term mean r ( t ) yRate of mean reversion k r ( t ) t
r (t ) = r (0) +
∫k
r
( t ) ( r ( t ) − r ( t ) )d t +
0
∫σ 0
FIRST Credit, Insurance & Risk
t
Credit Modelling
r
r ( t ) d Wr ( t )
Risk Risk Free Free Term Term Structure Structure (II) (II) σ σ
page 17
r
= 0 .1 , k r = 1 0 = 0 .1 , k r = 2
σ r = 0 .1 , k r = 2 σ r = 0 .2 , k r = 2
FIRST Credit, Insurance & Risk
r
Credit Modelling
Credit Credit Spread Spread Term Term Structure Structure (I) (I) ~
λ ( t ) = ρr ( t ) + x ( t ) determines correlation ~ between λ ( t ) and r ( t ) page 18
corr ( r ( t ), x ( t )) = 0
- Uncorrelated with interest rates - Continuous and jump component
yRandom jump size z, exponentially distributed θ e − θz , z > 0 yRandom number of jumps - follows Poisson process
FIRST Credit, Insurance & Risk
e − λ τ ( λτ ) n / n ! n = 0 ,1, 2 , ... τ = tim e interval Credit Modelling
Credit Credit Spread Spread Term Term Structure Structure (II) (II) yCredit spread component uncorrelated with the risk free interest rate t
x (t ) = x (0) + ∫ k x ( s)( x ( s) − x ( s))ds page 19
t
0
+ ∫ σ x (s) x (s)dWx (s) + 0
FIRST Credit, Insurance & Risk
Credit Modelling
Z (i ) ∑ τ
i ; (i ) ≤ t
Credit Credit Spread Spread Term Term Structure Structure (III) (III)
page 20
FIRST Credit, Insurance & Risk
more frequent and larger jumps
Credit Modelling
page 21
Part IV Credit Migration Model
FIRST Credit, Insurance & Risk
Credit Modelling
Credit Credit Migration Migration Model Model - Jumps modelled as changes in credit ratings and defaults - Continuous part modelled as continually changing risk premia
- Model jointly assets in various credit classes - Portfolio management and risk analysis
page 22
FIRST Credit, Insurance & Risk
- Calibration - incorporate economic and historical information
- Flexible in terms of data requirements and number of states
Credit Modelling
Markov Markov Chains Chains -- Generator Generator Matrix Matrix (I) (I) yContinuous time Markov chain yDiscrete state space 1
page 23
FIRST Credit, Insurance & Risk
2
~ 1 ⎛ λ1 ⎜ ~ 2 ⎜ λ21 ~ ⎜ . Λ= ⎜~ K-1 ⎜ λ K −1,1 ⎜ K ⎝ 0
~ λ12 ~
λ2
~
.
λ K −1,2 0
K-1
. . . . .
~ λ1, K −1 ~
λ 2 , K −1 . ~
λ K −1 0
constant over time absorbing state (default)
Credit Modelling
K
~ λ1K ⎞ ~ ⎟ λ2 K ⎟ . ⎟ ⎟ ~ λ K −1, K ⎟ ⎟ 0 ⎠
Markov Markov Chains Chains -- Generator Generator Matrix Matrix (II) (II) ~ I + Λ dt , transition matrix over short period dt y y
λij ≥ 0 , non-negative transition probabilities ~
page 24
y λi = − K
~
∑λ j ≥i
FIRST Credit, Insurance & Risk
ij
K
∑λ
~
i =1 j ≠i
≤
ij
, sum of all probabilities equals 1
~
∑λ j≥k
i + 1, j
, ∀ i, k k ≠ i + 1
A state i+1 is always more risky than state i Credit Modelling
Markov Markov Chains Chains -- Transition Transition Matrix Matrix yTransition matrix for the period t to T yExplicit computation ~ Λ = Σ −1 D Σ
~ Q ( t , T ) = Σ − 1 exp[ D ( T − t ) ]Σ page 25
FIRST Credit, Insurance & Risk
⎛ q~1 ( t , T ) ⎜ ~ ⎜ q21 (t , T ) ~ . Q (t , T ) = ⎜ ⎜~ ⎜ q K −1,1 ( t , T ) ⎜ 0 ⎝
q~1, K −1 ( t , T ) q~1K ( t , T ) ⎞ ⎟ q~2 , K −1 (t , T ) q~2 K ( t , T ) ⎟ ⎟ . . . ⎟ ~ ~ . q K −1 (t , T ) q K −1, K (t , T )⎟ ⎟ 0 1 . ⎠
. .
Credit Modelling
Model Model Structure Structure (I) (I) yStates : uniquely determine default probability yCredit ratings - can incorporate past credit rating transitions - non-Markovian model
~ Λ - Risk neutral generator matrix page 26
~ Λ - constant FIRST Credit, Insurance & Risk
jump in credit spread due to downgrade Credit Modelling
Model Model Structure Structure (II) (II) yIncorporate stochastic risk premia ~ ~ Λ stochastic = Λ × U ( t ) continuous process for risk premia page 27
FIRST Credit, Insurance & Risk
~ Λ
stochastic jump in credit spread due to downgrade Credit Modelling
Stochastic Stochastic Generator Generator Matrix Matrix yStochastic generator matrix arises from randomly changing risk premia ~ ~ Λ stochastic = Λ × U ( t ) Stochastic component page 28
t
t
0
0
U ( t ) = U ( 0 ) + ∫ ( a − kU ( t ) )dt + ∫ σ U ( t ) dWt Mean reverting process
FIRST Credit, Insurance & Risk
yClosed form formulae for bond prices Credit Modelling
Stochastic Stochastic Risk Risk Premia Premia y If eigenvectors are constant, can pose Λ ( t , T ) = Σ −1D ( t ) Σ
y Possible evolution of eigenvalues dX j = ( a j − b j X j ) dt + σ j dw ,
D ( t ) = diag( X j ( t )) Kj = 1
y Pricing equation is now modified to page 29
K
T ⎡ ⎛ ⎜ q iK ( t , T ) = ∑ ( Σ ) ij E exp ⎝ ∫ X j ( s ) ds⎞⎟⎠ D ( t ) ⎤ Σ jK ⎢⎣ ⎥⎦ t j =1 −1
y Expectation has closed (algebraic) form FIRST Credit, Insurance & Risk
− depends on parameters a b σ and on D(t)
Credit Modelling
Calibration Calibration (I) (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 i ( 0, T )
~ Λ stochastic (Λ , a , k ,σ )
page 30
Simulate Simulate Credit CreditSpread Spread
Λ FIRST Credit, Insurance & Risk
- Historical generator matrix (estimated from one year transition matrix) - Credit spread historical time series
Credit Modelling
Price exotic structures
Calibration Calibration (II) (II) yLeast squares fit to match directly observed coupon bond prices (any number)
(
page 31
)
2 ~ ⎧ K Ji 2 ⎞⎫ T K ⎛ λ −λ ⎞ ⎛ ij ⎪ ~ ⎜ ij ⎟⎪ i i i + min P − F ( h ) v ( h ; Λ ) ⎟ ⎜ ⎬ ∑ ∑ j j ~ ⎨∑ ∑ ⎜ ⎟ Λ ⎠ ⎝ β h =1 i , j =1 ij ⎪ i =1 j =1 ⎝ ⎠ ⎪⎭ ⎩ coupon at market price of date h coupon bond for confidence prior class i level generator matrix
yObtain solution closest to the historical generator
FIRST Credit, Insurance & Risk
matrix Λ - stable calibration
Credit Modelling
Calibration Calibration -- Emerging Emerging Markets Markets (II) (II) 500 450 400 350 3 states
page 32
300
5 states
250
Actual
200 150 100
Credit, Insurance & Risk
Credit Modelling
30
28
26
24
22
20
18
16
14
12
10
8
6
4
FIRST
2
0
0
50
Calibration Calibration -- Corporate Corporate Market Market (II) (II) 1400 1200 AAA AA
page 33
Credit spread (bp)
1000
A BBB
800
BB 600
B CCC
400 200
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Maturity (years)
Credit Modelling
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
Calibration Calibration -- US US Industrials Industrials (I) (I)
page 34
Credit Spread (bp’s)
180 160
AAA
140
AA A
120 100 80 60 40 20
Credit, Insurance & Risk
Maturity (years) Credit Modelling
30
28
26
24
22
20
18
16
14
12
8
6
4
10
FIRST
2
0
0
Calibration Calibration -- US US Industrials Industrials (II) (II) 1200 BBB
page 35
Credit Spread (bp’s)
1000
BB B
800
CCC
600
400
200
Credit, Insurance & Risk
Maturity (years) Credit Modelling
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
FIRST
0
0
page 36
Part V Estimating Credit Spread Volatility
FIRST Credit, Insurance & Risk
Credit Modelling
Credit Credit spread spread volatilities volatilities estimates estimates yy74 74Bonds Bonds yy67 67Investment Investmentgrade grade(Baa (Baaand andabove) above)US US Industrial Industrialbonds bonds yy77Speculative Speculativegrade grade(Ba (Baand andbelow) below) Emerging EmergingMarket Marketbonds bonds
yy88Model Modelstates states page 37
FIRST Credit, Insurance & Risk
yy 11Moody’s Moody’sAaa Aaa yy 22Moody’s Moody’sAa1 Aa1--Aa3 Aa3 yy 33Moody’s Moody’sA1 A1--A3 A3 yy 44Moody’s Moody’sBaa1 Baa1--Baa3 Baa3
55Moody’s Moody’sBa1 Ba1--Ba3 Ba3 66Moody’s Moody’sB1 B1--B3 B3 77Moody’s Moody’sCCC CCC 88Default Default
yyHistorical Historicalgenerator generatormatrix matrixfrom fromMoody’s Moody’s average average1y 1ytransition transitionmatrix matrix1920 1920--1996 1996 Credit Modelling
Estimated Estimated short short term term spreads spreads 0.2
0.15
Credit ratings
0.1
0.05
0
0
50
100
150
200
AA
250
A
BBB BB B CCC page 38
Lower ratings have − higher spread − higher volatility
FIRST Credit, Insurance & Risk
Credit Modelling
Eigenvalues Eigenvalues of of generator generator matrix matrix Principal components
Eigenvalues 0
0.05
-0.05
0 -0 05 0 0.05
-0.1
-0.05 0 0.05
-0.2
-0.05 0 0.05
-0.3
Credit, Insurance & Risk
200
250
50
100
150
200
250
50
100
150
200
250
50
100
150
200
250
0
0
50
100
150
200
250 -0.05 0
FIRST
150
0
-0.25
-0.35
100
0
-0.15
page 39
50
- first 3 principal components account for 99% of the variance Credit Modelling
Advanced Advanced Modelling Modelling Issues Issues Stochastic Recovery Rates - Recovery rates are random with high variance - Exogenous - Endogenous - depend on the severity of default
Credit Events Correlated with Interest Rates - Credit migration and defaults depend on interest rates - Joint state variables for interest rates and credit spreads - Incorporate business cycles
page 40
Non - Markovian Bankruptcy Process
FIRST Credit, Insurance & Risk
- Autocorrelated migration process - Markovian in state space augmented with lagged values
Second Generation Products - Basket options - credit spread, default correlations - Multiple Currencies - Quantos
Credit Modelling
