Copyright © John Hull 2008 1 Dynamic Models of Portfolio Credit Risk: A Simplified Approach John Hull Princeton Credit Conference May 2008
Jan 20, 2016
Copyright © John Hull 2008 1
Dynamic Models of Portfolio Credit Risk:
A Simplified Approach
John Hull
Princeton Credit Conference
May 2008
7th Edition Out!
Copyright © John Hull 2008 2
Copyright © John Hull 2008 3
January 30, 2007 DataTable 1
iTraxx CDO tranche quotes January 30, 2007.
aL aH 3 yr 5 yr 7 yr 10 yr
0 0.03 n/a 10.25% 24.25% 39.30%0.03 0.06 n/a 42.00 106.00 316.000.06 0.09 n/a 12.00 31.50 82.000.09 0.12 n/a 5.50 14.50 38.250.12 0.22 n/a 2.00 5.00 13.75
Index 15.00 23.00 31.00 42.00
CDX IG CDO tranche quotes January 30, 2007.
aL aH 3 yr 5 yr 7 yr 10 yr
0 0.03 n/a 19.63% 38.28% 50.53%0.03 0.07 n/a 63.00 172.25 427.000.07 0.10 n/a 12.00 33.75 96.000.10 0.15 n/a 4.50 14.50 43.250.15 0.30 n/a 2.00 6.00 13.75
Index 19.00 31.00 43.00 56.00
Copyright © John Hull 2008 4
CDO Models
Standard market model is one-factor Gaussian copula model of time to default
Alternatives that have been proposed: t-, double-t, Clayton, Archimedian, Marshall Olkin, implied copula
All are static models. They provide a probability distribution for the loss over the life of the model, but do not describe how the loss evolves
Dynamic models are needed to value options and structured deals such as leveraged super seniors
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Dynamic Models for Portfolio Losses: Prior Research
Structural: Albanese et al; Baxter (2006); Hull et al (2005)
Reduced Form: Duffie and Gârleanu (2001), Chapovsky et al (2006), Graziano and Rogers (2005), Hurd and Kuznetsov (2005), and Joshi and Stacey (2006)
Top Down: Sidenius et al (2004), Bennani (2005), Schonbucher (2005), Errais, Giesecke, and Goldberg (2006), Longstaff and Rajan (2006), Putyatin et al (2005), and Walker (2007)
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Our Objective
Build a simple dynamic model of the evolution of losses that is easy to implement and easy to calibrate to market data
The model is developed as a reduced form model, but can also be presented as a top down model
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Specific vs General Models
Specific models track the evolution of default risk on a single name or portfolio that remains fixed (e.g. describes how credit spread for a particular company evolves)
General models track the evolution of default risk on a single name or portfolio that is updated so that it always has certain properties (e.g. describes how the average spread for an A-rated company evolves)
We focus on specific models
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CDO Valuation Key to valuing a CDO lies in the calculation of expected tranche
principal on payment dates Expected payment on a payment date equals spread times
expected tranche principal on that date Expected payoff between payment dates equals reduction in
expected tranche principal between the dates Expected accrual payments can be calculated from expected
payoffs Expected tranche principal at time t can be calculated from the
cumulative default probabilities up to time t and recovery rates of companies in the portfolio
The Model
Instead of modeling the hazard rate, h(t) we model
This is –ln[S(t)] where S(t) is the survival probability calculated from the path followed by the hazard rate between times 0 and t
Filtration: We assume that at time t we know the path followed by S between time zero and time t and the number of defaults up to time t
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t
dhX0
)(
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The Model (Homogeneous Case)
where dq represents a jump that has intensity and jump size H
and are functions only of time and H is a function of the
number of jumps so far. > 0, H > 0.
dqdtdX
X + t
X + t + Ht
tX
The Hazard Rate Process
The hazard rate process is
where dI is an impulse that has intensity
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dIdttdh )(
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Analytic Results and Binomial Trees
Once (t), (t), and the the size of the jth jump, Hj, have been specified the model can be used to value analytically CDOs Forward CDOs Options on CDOs
For other instruments a binomial tree can be used
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Binomial Tree for Model
))((1
))((
)(
1
1
0
iii
iii
i
i
i
dttM
movement down ofy Probabilit
time at movement up ofy Probabilit
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Simplest Version of Model
Jump size is constant and (t), is zero Jump intensity, (t) is chosen to match the
term structure of CDS spreads There is then a one-to-one correspondence
between tranche quotes and jump size Implied jump sizes are similar to implied
correlations
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Comparison of Implied Jump Sizes with Implied Tranche Correlations
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Possible Extensions to Fit All Market Data
Multiple processes each with its own jump size and intensity
Intensity and jump size changing in different intervals: 0 to 5 yrs, 5 to 7 yrs, and 7 to 10 yrs
Model can (in principle) fit all quotes simultaneously
We have chosen to focus on extensions where there are relatively few parameters
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Extension Involving Three Parameters
The size of the Jth jump is HJ = H0 eJ
The three parameters are The initial jump size The growth in the jump size The jump intensity
The model reflects empirical research showing that correlation is higher in adverse market conditions
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Variation of best fit jump parameters,
H0 and across time. (10-day moving
averages)
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Variation of jump intensity (10-day moving averages)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
4-Jul-06 3-Aug-06 2-Sep-06 2-Oct-06 1-Nov-06 1-Dec-06 31-Dec-06
Jump Intensity,
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Evolution of Loss Distribution on January 30, 2007 for 3-parameter model.
0.00
0.10
0.20
0.30
0% 3% 6% 9% 12% 15% 18% 21% 24%
Loss (% of Total Notional)
Unconditional Loss Distribution at 4 Maturities
3-Years 5-Years 7-Years 10-Years
Probabilities for losses greater than 9% multiplied by 100.
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Valuation of Forward Contracts on CDOs that End in Five Years Using 3-Parameter Model on January 30, 2007
Table 4
Breakeven tranche spread for forward start CDO tranches on iTraxx on January 30, 2007. The tranches mature in five years. Results are based on the three-parameter model in Section IIIC calibrated to the market data in Table 1.
Tranche Start 1.0 2.0 3.0 4.0 4.5
aL aH Breakeven Tranche Spreads
0 0.03 11.5 11.3 11.1 6.4 3.4
0.03 0.06 54.0 70.1 93.2 124.4 144.2
0.06 0.09 14.7 19.4 26.1 35.2 40.7
0.09 0.12 5.8 7.7 10.6 14.8 17.5
0.12 0.22 2.0 2.6 3.7 5.3 6.3
Index 25.3 29.1 36.7 41.4 43.7
Copyright © John Hull 2008 22
Valuation of At-The-Money Options (in basis points) on CDOs that End in Five Years Using 3-Parameter Model on January 30, 2007
Table 5
Prices in basis points of at-the-money European options on iTraxx CDO tranches on January 30, 2007. The tranches mature in five years. Results are based on the three-parameter model in Section IIIC calibrated to the market data in Table 1.
Option Expiry in Years
aL aH 1.0 2.0 3.0 4.0 4.5
0.03 0.06 67.8 91.3 89.7 68.3 41.4
0.06 0.09 23.2 29.8 30.7 23.1 13.5
0.09 0.12 9.7 12.2 13.3 10.0 6.1
0.12 0.22 3.7 4.4 5.0 3.8 2.4
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Implied Volatilities from Black’s Model as Option Maturity is Changed
Table 6
Implied volatilities of at-the-money European style options on iTraxx CDO tranches on January 30, 2007. The tranches mature in five years. Results are based on the three-parameter model in Section IIIC calibrated to the market data in Table 1.
Option Expiry in Years
aL aH 1.0 2.0 3.0 4.0 4.5
0.03 0.06 96.1% 100.9% 96.8% 104.3% 107.5%
0.06 0.09 123.2% 122.6% 125.6% 137.2% 135.3%
0.09 0.12 133.0% 128.6% 139.4% 144.8% 148.9%
0.12 0.22 149.3% 137.3% 160.5% 160.6% 181.8%
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Implied Volatilities for 2 yr Option from Black’s Model as Strike Price is Changed
Table 7
Implied volatilities for 2-year European options on iTraxx CDO tranches for strike prices between 75% and 125% of the forward spread on January 30, 2007. The tranches mature in five years. Results are based on the three-parameter model in Section IIIC calibrated to the market data in Table 1.
CDO Tranche
K / F 3 to 6% 6 to 9% 9 to 12% 12 to 22%
0.75 100.1% 125.6% 132.9% 143.5%
0.80 100.6% 125.2% 132.1% 142.3%
0.85 100.9% 124.7% 131.3% 141.1%
0.90 101.1% 124.1% 130.5% 139.9%
0.95 101.0% 123.4% 129.5% 138.6%
1.00 100.9% 122.6% 128.6% 137.3%
1.05 100.6% 121.7% 127.5% 136.0%
1.10 100.3% 120.9% 126.5% 135.2%
1.15 99.8% 119.9% 125.4% 135.3%
1.20 99.3% 119.0% 125.0% 136.0%
1.25 98.8% 118.0% 124.8% 137.1%
Copyright © John Hull 2008 25
Leverage Super Senior with Loss Trigger
Total exposure of seller of protection is limited to a fraction x of the tranche notional
When losses reach some level the buyer of protection can cancel the deal and seller has to pay the value of the tranche to the buyer. Define n* as the number of losses triggering cancellation
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Breakeven LSS spread on Jan 30, 2007 as a function of the maximum percentage loss by protection seller, x%, and the number of defaults triggering close out, n*
0.5
0.7
0.9
1.1
1.3
1.5
0 2 4 6 8 10 12 14 16
Number of Defaults Required to Trigger Termination
x = 0.05 x = 0.10 x = 0.20
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Extensions
Model can be extended so that Different companies have different CDS spreads The recovery rate is negatively correlated with the
default rate iTraxx and CDX jumps are modeled jointly
Bespokes
Calibrate homogeneous model to iTraxx and CDX IG
If all names are North American, use a non-homogeneous model where underlying companies have the the CDX IG jumps and their own drifts.
If there are a mixture of European and North American names use a non-homogeneous model where the iTraxx and CDX IG jumps are modeled jointly
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Conclusions
It is possible to develop a simple dynamic model for losses on a portfolio by modeling the cumulative default probability for a representative company
The only way of fitting the market appears to be by assuming that there is a small probability of a series of progressively bigger jumps in the cumulative probability.
As credit market deteriorates default correlation becomes higher