Copyright © John Hull 2008 1 Dynamic Models of Portfolio Credit Risk: A Simplified Approach John Hull Princeton Credit Conference May 2008

Copyright © John Hull 2008 1

Dynamic Models of Portfolio Credit Risk:

A Simplified Approach

John Hull

Princeton Credit Conference

May 2008

7th Edition Out!



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


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


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)


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


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


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


t

dhX0

)(


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


dIdttdh )(


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


Binomial Tree for Model

))((1

))((

)(

1

1

0

iii

iii

i

i

i

dttM

movement down ofy Probabilit

time at movement up ofy Probabilit


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


Comparison of Implied Jump Sizes with Implied Tranche Correlations


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


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


Variation of best fit jump parameters,

H0 and across time. (10-day moving

averages)


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,


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.


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


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


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%


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%


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


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


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



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

Documents

Copyright © John Hull 2008 1 Dynamic Models of Portfolio Credit Risk: A Simplified Approach John Hull Princeton Credit Conference May 2008