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JUMP DRIVEN PREPAYMENT AND DEFAULT MODELS FOR LCDS, ABS AND PORTFOLIOS OF LCDSs Wim Schoutens - K.U.Leuven - wim@schoutens.be Wim Schoutens, 09-09-2008 Linz, Austria - p. 1/35 Joint Work Joao Garcia (Dexia Holding) Serge Goossens (Dexia

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SLIDE 1

Wim Schoutens, 09-09-2008 Linz, Austria - p. 1/35

JUMP DRIVEN PREPAYMENT AND DEFAULT MODELS FOR LCDS, ABS AND PORTFOLIOS OF LCDSs

Wim Schoutens - K.U.Leuven - wim@schoutens.be

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SLIDE 2

Outline

  • Joint Work
  • Content

Introduction ABS Modeling LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 2/35

Joint Work

■ Joao Garcia (Dexia Holding) ■ Serge Goossens (Dexia Bank) ■ Hansjörg Albrecher (TU Graz) ■ Sophie Ladoucette (Secura Re) ■ Victoriya Masol (EURANDOM) ■ Peter Dobranszky (K.U.Leuven - Finalyse) ■ Geert Van Damme (K.U.Leuven) ■ Marcella Belluci (EIB)

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SLIDE 3

Outline

  • Joint Work
  • Content

Introduction ABS Modeling LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 3/35

Content

■ Introduction ◆ Basic of jump processes ■ ABS Modeling ◆ Default Fraction Models ◆ Prepayment Fraction Models ◆ Impact on Rating and WAL ■ LCDX Modeling ◆ LCDS implied default and prepayment ◆ Joint modeling of prepayment and default ◆ LCDX one factor model ◆ LCDX base correlation

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SLIDE 4

Outline Introduction

  • Why Lévy ?
  • What is a Lévy process ?
  • Defining a Lévy process
  • Examples of Lévy : Gamma

ABS Modeling LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 4/35

Why Lévy ?

■ Credit market events are very shock driven. ■ Jumps and heavy tails are important features in the modeling.

2.5 3 3.5 4 4.5 5 0.05 0.1 0.15 0.2 0.25 Gamma tail (a=3, b=2) versus Gaussian tail with same mean and variance x f(x) Gamma tail Gaussian tail

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SLIDE 5

Outline Introduction

  • Why Lévy ?
  • What is a Lévy process ?
  • Defining a Lévy process
  • Examples of Lévy : Gamma

ABS Modeling LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 5/35

What is a Lévy process ?

■ Lévy processes are generalization of Brownian Motions (stationary and

independent increments) allowing for

◆ non-Gaussian underlying distribution (skewness, kurtosis, more heavier

tails, ...);

◆ jumps.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Standard Brownian Motion 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.15 −0.1 −0.05 0.05 0.1 VG Process (C=20; G=40; M=50)

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SLIDE 6

Outline Introduction

  • Why Lévy ?
  • What is a Lévy process ?
  • Defining a Lévy process
  • Examples of Lévy : Gamma

ABS Modeling LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 6/35

Defining a Lévy process

■ Formal definition: X = {Xt, t ≥ 0} is a Lévy process if ◆ X0 = 0 ◆ X has independent increments ◆ X has stationary increments ◆ the increment Xt+s − Xt follows a infinitely divisible distribution. ■ Examples of infinitely divisible distributions: ◆ Normal → Brownian motion ◆ Poisson → Poisson process ◆ Gamma → Gamma process ◆ IG → IG process ◆ CMY → CMY process ◆ Variance Gamma → VG process ◆ Normal Inverse Gaussian → NIG process ◆ Meixner → Meixner process ◆ Generalized Hyperbolic → GH process ◆ etc.

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SLIDE 7

Outline Introduction

  • Why Lévy ?
  • What is a Lévy process ?
  • Defining a Lévy process
  • Examples of Lévy : Gamma

ABS Modeling LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 7/35

Examples of Lévy : Gamma

■ The density function of the Gamma distribution Gamma(a, b) with

parameters a > 0 and b > 0 is given by: fGamma(x; a, b) = ba Γ(a) xa−1 exp(−xb), x > 0.

■ The Gamma-process G = {Gt, t ≥ 0} with parameters a, b > 0 is a

stochastic process which starts at zero and has stationary, independent Gamma-distributed increments and Gt follows a Gamma(at, b) distribution.

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 Gamma Process − ν = 0.15

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SLIDE 8

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 8/35

Model Overview

■ We model the default and prepayment fraction of ABS of a certain class of

loans which can prepay and default.

■ Default Models: ◆ logistic ◆ Cox (Gamma) ◆ one-factor (Gaussian and Lévy) ■ Prepayment Models: ◆ Constant Prepayment Rate ◆ Cox (Gamma) ◆ one-factor (Gaussian and Lévy)

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SLIDE 9

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 9/35

Logistic Default Model

■ The Logistic curve

F(t) = a 1 + be−c(t−t0) ,

■ a has to be chosen from a predetermined default distribution, e.g. the

Log-Normal distribution with mean µ and standard deviation σ.

■ Each different value for a will give rise to a new default curve.

20 40 60 80 100 120 0.1 0.2 0.3 0.4 0.5 t F(t) Logistic default curves (µ = 0.20 , σ = 0.10) a = 0.4133 a = 0.3679 a = 0.2234 a = 0.1047 a = 0.0804 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 fX X ∼ LogN(µ, σ) Probability density of LogN(µ = 0.20 , σ = 0.10)

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SLIDE 10

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 10/35

Cox Default Models

■ Based on a on a single sided Lévy process X = {Xt, t ≥ 0}. ■ The fraction of loans that have defaulted (prepaid) at time t :

1 − exp(−Xt).

■ Hence the fraction of loans that have not defaulted (prepaid) equals

exp(−Xt).

■ ∆t time later the number of loans that by then have not defaulted (prepaid)

equals exp(−Xt+∆t).

■ Hence

(1 − exp(−Xt+∆t)) − (1 − exp(−Xt)) exp(−Xt) ≈ log(exp(−Xt)) − log(exp(−Xt+∆t)) = Xt+∆t − Xt.

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SLIDE 11

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 11/35

Cox Gamma Default Model

■ Based on a Gamma Process G = {Gt, t ≥ 0}: ■ The percentage number of new defaults (prepayments) in each period of

length ∆t is Gamma(a∆t, b) distributed.

■ One can match a and b to a preset mean number and variance of defaults

at maturity: E[1 − exp(−GT )] = 1 − (1 + 1/b)−aT = µG(T); V ar[1 − exp(−GT )] = (1 + 2/b)−aT − (1 + 1/b)−2aT = σ2

G(T).

20 40 60 80 100 120 0.1 0.2 0.3 0.4 0.5 t Pd(t) Cox default curve (µ = 0.20 , σ = 0.10) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.1 0.2 0.3 0.4 0.5 X ∼ 1−e−λ(T) fX Probability density of 1−e−λ(T), with λ(T) ∼ Gamma(aT = 2.99, b = 12.90)

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SLIDE 12

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 12/35

Gaussian Copula Model

■ The Gaussian one-factor model (Vasicek, Li) assumes the following

"dynamics" for the standardized firm’s latent value:

◆ Ai(T) = √ρ Y + √1 − ρ ǫi, i = 1, . . . , n; ◆ where Y and ǫi, i = 1, . . . , n are i.i.d. standard normal variables. ■ The ith obligor defaults at time T if Ai(T) falls below some preset barrier

Ki(T) (for CDOs extracted from CDS quotes to match individual default probabilities).

■ This model is actual based on the Multivariate Normal / Gaussian Copula

with its known problems (cfr. correlation smile in CDO models).

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SLIDE 13

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 13/35

Tails

■ The underlying reason is the too light tail-behavior of the standard normal

rv’s (a large number of joint defaults will be caused by a very negative common factor Y ): Ai(T) = √ρ Y +

p

1 − ρ ǫi, i = 1, . . . , n.

■ One has to pump up artificially the (base) correlation (e.g. in order to get

market observed CDO prices for the senior tranches).

■ Therefore we look for models where the distribution of the factors has

more heavy tails than the normal distribution.

■ Different tail classes: ◆ light tails : Normal distribution ◆ semi-heavy tails : Exp, Gamma, VG, ... ◆ heavy tails (EVT theory): stable, ... ■ How to set up a multivariate exponentially tailed model ?

Ai(T) has exponential tail and Corr[AiAj] = ρ, i = j

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SLIDE 14

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 14/35

Correlation via Running Time

■ Write the "dynamics" for the firm’s latent value:

Ai(T) = √ρ Y +

p

1 − ρ ǫi, i = 1, . . . , n, where Y and ǫi, i = 1, . . . , n are i.i.d. standard normal variables: Ai(T) = Wρ + W (i)

1−ρ, i = 1, . . . , n,

where W and the W (i)’s are independent standard Brownian motions

■ Idea: correlate by letting Brownian processes run some time together

and then let them free (independence). Their end-values (at t = 1) are standard multinormal with correlation ρ.

0.2 0.4 0.6 0.8 1 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 time Correlated outcomes ρ=0.3 0.2 0.4 0.6 0.8 1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 Correlated outcomes ρ=0.8 time

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SLIDE 15

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 15/35

Idea works for every Lévy Processes

■ Let X = {Xt, t ∈ [0, 1]} be a Lévy process based on an infinitely divisible

distribution: X1 ∼ L.

■ Denote the cdf of Xt by Ht(x), t ∈ [0, 1], and assume it is continuous:

P(Xt ≤ x) = Ht(x).

■ Let X = {Xt, t ∈ [0, 1]} and X(i) = {X(i)

t , t ∈ [0, 1]}, i = 1, 2, . . . , n be

independent and identically distributed Lévy processes

■ All processes are independent of each other and are based on the

same mother infinitely divisible distribution L.

■ Let 0 < ρ < 1, be the correlation that we assume between the defaults

  • f the obligors.
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SLIDE 16

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 16/35

Lévy Copula Model

■ We propose the generic one-factor Lévy model. ■ We assume that the asset value of obligor i = 1, . . . , n is of the form:

Ai(T) = Xρ + X(i)

1−ρ,

i = 1, . . . , n.

■ Each Ai = Ai(T) has by the stationary and independent increments

property the same distribution as the mother distribution L with distribution function H1(x).

■ Further we have that if E[X2

1] < ∞

Corr[Ai, Aj] = E[AiAj] − E[Ai]E[Aj]

p

Var[Ai]

p

Var[Aj] = ρ.

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SLIDE 17

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 17/35

Gaussian Copula Model : Example

■ One could choose barrier such that the average probability of default

before time t equals some prespecified level, determined by a Poisson process with constant intensity λ, i.e. Pd(t) = Pr

h

Zi ≤ Hd

t

i

= Φ

h

Hd

t

i

= Pr [Nt > 0] = 1 − e−λt, where λ is set such that Pd(T) = µ, with µ the predetermined value for the mean of the default distribution; ρ is set to match the Variance of the default distribution at time T.

20 40 60 80 100 120 0.1 0.2 0.3 0.4 0.5 t Pd(t) Normal 1−factor default curve (µ = 0.20 , σ = 0.10) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.1 0.2 0.3 0.4 0.5 #{Zi ≤ HT} f#{Z

i ≤ H T}

Probability density of the cumulative default rate at time T (ρ = 0.12135)

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SLIDE 18

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 18/35

Constant Prepayment Rate

■ The standard model (100% PSA) works as follows: starting with an

annualized prepayment rate of 0% in month 0, the rate increases by 0.2% each month, until it peaks at 6% after 30 months. From the 30th month on, the model assumes an annual constant prepayment rate of 6%. Some times

  • ne also refers to this model as 100% PSA.

■ A more general two parameter model: ◆ The first parameter τ0 represents the point in time where one switches

from an linear increasing rate into a constant rate (for the CPR, τ0 = 2.5 year).

◆ A second parameter represents the fraction of loans that have prepaid

at the maturity T: pp(T). A value of pp(T) = 0.20 means that at time T 20 percent of the loans have prepaid.

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SLIDE 19

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 19/35

Constant Prepayment Rate

■ Once these parameters are set, one can calculate how much the rate

increase per time unit before τ0 should be and at which level it will remail constant thereafter. The slope of the prepayment rate until τ0 is given by: a = pp(T)/(τ 2

0 /2 + (T − τ0)τ0).

■ The prepayment curve (the integrated prepayment rate) reads

pp(t) = at2/2, 0 ≤ t ≤ τ0 = aτ 2

0 /2 + aτ0(t − τ0),

τ0 ≤ t ≤ T.

1 2 3 4 5 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Contstant Prepayment Rate Model (pp(5)=0.2, τ0=1.5 y) prepayment fraction/probability prepayment rate τ0 pp(T) a

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SLIDE 20

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 20/35

Cox prepayment Model

■ The Cox prepayment model is completely analogous to the Cox default

model, with Pd(t) replaced by Pp(t), i.e. the cumulative prepayment rate at time t.

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SLIDE 21

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 21/35

1-factor prepayment models

■ The 1-factor prepayment model starts from the same underlying

philosophy as its default equivalent.

■ The barrier Hp

t is chosen such that the average probability of prepayment

before time t equals CPR(t): Pp(t) = Pr [Zi ≥ Hp

t ] = 1 − Φ [Hp t ] = CPR(t),

which implies, Hp

t = Φ−1 [1 − CPR(t)] ,

with Φ the cumulative distribution function of the underlying distribution.

20 40 60 80 100 120 0.1 0.2 0.3 0.4 0.5 t Pp(t) Normal 1−factor prepayment curve (µ = 0.20 , σ = 0.10) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.1 0.2 0.3 0.4 0.5 #{Zi ≥ HT} f#{Z

i ≥ H T}

Probability density of the cumulative prepayment rate at time T (ρ = 0.12135)

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SLIDE 22

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 22/35

ABS Modeling

■ One can now build these default and prepayment models into any scenario

generator for pricing, rating and determining the Weighted Average Life (WAL) of notes backed by an underlying asset pool.

■ Many combination of the above described default and prepayment models

are possible.

■ We illustrate the use of the models on a simple waterfall structure.

ASSETS Initial balance of the asset pool V0 $30e+06 Number of loans in the asset pool N0 2000 Weighted Average Maturity of the assets WAM 10 years Weighted Average Coupon of the assets WAC 12%/year Payment frequency monthly Reserve target 5% Eligible reinvestment rate 3.85%/year Loss-Given-Default LGD 50% Lag 5 months

LIABILITIES Initial balance of the senior note A0 $24e+06 Premium of the senior note rA 7%/year Initial balance of the subordinated note B0 $6e+06 Premium of the subordinated note rA 9%/year Servicing fee rsf 1%/year Servicing fee shortfall rate rsf−sh 20%/year Payment method Pro-rata Sequential

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SLIDE 23

Outline Introduction ABS Modeling

  • Model Overview
  • Logistic Default Model
  • Cox Default Models
  • Cox Gamma Default Model
  • Gaussian Copula Model
  • Tails
  • Correlation via Running Time
  • Idea works for every Lévy

Processes

  • Lévy Copula Model
  • Gaussian Copula Model :

Example

  • Constant Prepayment Rate
  • Constant Prepayment Rate
  • Cox prepayment Model
  • 1-factor prepayment models
  • ABS Modeling
  • ABS Modeling Example

LCDX Modeling Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 23/35

ABS Modeling Example

■ ABS note: µd = 0.20, σd = 0.10, µp = 0.20 and σp = 0.10 (pro rata; with

reserve account).

Prepayment models CPR Cox Normal 1-factor A B A B A B Default models Logistic DIRR (bps) 3.3503e-01 1.1454e+01 3.8263e-01 1.3032e+01 4.0013e-01 1.1898e+01 Moody’s Aa1 A3 Aa1 Baa1 Aa1 A3 Pr.Def. (%) 0.68 3.98 0.77 4.69 0.88 4.03 S&P’s BBB- BB- BB+ BB- BB+ BB- WAL (yr) 5.27 5.31 5.13 5.17 5.27 5.31 Cox DIRR (bps) 1.7790e-01 1.8216e+01 3.1961e-01 2.1471e+01 2.0203e-01 1.8269e+01 Moody’s Aa1 Baa1 Aa1 Baa2 Aa1 Baa1 Pr.Def. (%) 0.80 19.92 1.24 21.66 1.22 21.00 S&P’s BB+ B BB B BB B WAL (yr) 5.26 5.36 5.11 5.22 5.26 5.36 Normal 1-factor DIRR (bps) 2.8147e-02 1.6229e+00 6.7938e-02 1.7039e+00 5.1950e-02 8.6092e-01 Moody’s Aaa A3 Aa1 A3 Aaa1 Aaa3 Pr.Def. (%) 0.10 0.43 0.13 0.53 0.06 0.37 S&P’s BBB+ BBB- BBB BBB- A- BBB- WAL (yr) 5.24 5.25 5.09 5.10 5.25 5.25 Lévy 1-factor DIRR (bps) 5.5959e+00 2.4214e+01 7.6642e+00 4.6924e+01 Moody’s A2 Baa2 A2 Baa1 Pr.Def. (%) 1.50 1.96 1.85 2.56 S&P’s BB BB BB BB- WAL (yr) 5.26 5.30 5.12 5.18

slide-24
SLIDE 24

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 24/35

LCDS

■ Loan Credit Default Swaps (LCDSs) are instruments that provide the buyer

an insurance against the defaulting of a loan.

■ They are very similar to Credit Default Swaps (CDSs), except that with

LCDSs there is a possibility that the loan prepays earlier.

1 2 3 4 5 140 160 180 200 220 240 260 280 300 320 maturity spread (bp) LCDS − ACS − 31−JAN−2008

slide-25
SLIDE 25

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 25/35

Implied Prepayment and Default Prob.

■ If both CDS and LCDS are available, one can in theory extract P (i)

default(t)

  • ut of CDS curve and fixing this extract P (i)

prepay(t) out of LCDS curve.

■ However not always CDS and LCDS are both available. Moreover, LCDS

does often not include restructuring into default definition.

■ A. Berndt, R.A. Jarrow and C. Kan (2007) Restructuring Risk in Credit

Default Swaps: An Empirical Analysis, Stochastic Processes and their Applications 117 (11):

”Comparing quotes from default swap (CDS) contracts with a restructuring event and without, we find that the average premium for restructuring risk represents 6% to 8%

  • f the swap rate without restructuring.”

■ Also any other (CPR, 100%PSA, Gamma prepayment model, ...) can be

used to fix prepayment process and P (i)

prepay(t). Then P (i) default(t) can be

extracted out of LCDS.

slide-26
SLIDE 26

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 26/35

LCDX Modeling

■ We propose to use the same generic one-factor Lévy model:

Ai(T) = Xρ + X(i)

1−ρ,

i = 1, . . . , m.

■ Again, the ith loan defaults before t if Ai is below some preset barrier Ki(t):

P (i)

default(t) = P(Ai ≤ Ki(t))

■ Moreover, the ith loan prepays before t if Ai is above some preset barrier

Hi(t): P (i)

prepay(t) = P(Ai ≥ Hi(t)).

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −3 −2 −1 1 2 3 loan I loan II loan III loan IV loan V prepayment barrier default barrier deafult time of loan IV prepayment time of loan III

slide-27
SLIDE 27

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 27/35

LCDX Tranche Pricing

■ Very similar as in the evaluation of a CDO tranche, we have

spread =

Pn

i=1(E[LT r i ] − E[LT r i−1])d(ti)

Pn

i=1(1 − E[LT r i ] − E[PP T r i

])d(ti)∆ti . where E[LT r

i ] is the expected Loss on tranche Tr and E[PP T r i

] is the expected prepayment on tranche Tr at time ti.

■ Note: if a default occurs, both the junior-most and the senior-most tranches

are impacted. The junior-most tranche notional is reduced by the amount

  • f the loss. The senior-most tranche notional is amortized by the amount
  • f the recovery.

■ In order to calculate expected tranche losses and expected outstanding

notional of the portfolio of m LCDSs at a point in time t, we need the joint probability of having k defaults and l prepayments upto time t: Πk,l(t) = P(k default and l prepayments until t) =

Z +∞

−∞

P(k default and l prepayments until t|Xρ = y)dHρ(y)

slide-28
SLIDE 28

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 28/35

Conditional Probabilities

■ Denote by pi(y; t) is the probability that the latent variable Ai is below the

default barrier Ki(t), given that the systematic factor Xρ takes the value y: pi(y; t) = P(Xρ + X(i)

1−ρ ≤ Ki(t))|Xρ = y)

= H1−ρ(Ki(t) − y)

■ Similarly, denote by qi(y; t) the probability that the latent variable Ai is

above the prepayment barrier Hi(t), given that the systematic factor Xρ takes the value y: qi(y; t) = P(Xρ + X(i)

1−ρ ≥ Hi(t))|Xρ = y)

= 1 − H1−ρ(Hi(t) − y).

slide-29
SLIDE 29

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 29/35

Recursive loss-prepayment formula

■ Denote by Πy,n

k,l (t) = P(k default and l prepayments until t|Xρ = y).

■ Then by the bivariate recursive loss-prepayment formula, we have

Πy,0

0,0(t)

= 1 Πy,n+1

k,l

(t) = (1 − pn+1(y; t) − qn+1(y; t))Πy,n

k,l (t)

+pn+1(y; t)Πy,n

k−1,l(t) + qn+1(y; t)Πy,n k,l−1(t)

where we assume Πy,n

−1,l(t) = Πy,n k,−1(t) = 0 for notational simplicity.

■ This leads to the joint unconditional probability to have k defaults and l

prepayments out of a group of m firms Πk,l(t) =

Z +∞

−∞

Πy,m

k,l dHρ(y).

slide-30
SLIDE 30

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 30/35

LCDX Base Correlation

■ Gaussian LCDX base correlation:

0.05 0.08 0.12 0.15 45 50 55 60 65 70 75 80 85 90 95 tranche base correlatiuon (%) Base Correlation LCDX.NA.9−V1 5Y 29−02−2008 31−03−2008 23−04−2008

slide-31
SLIDE 31

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 31/35

LCDX VG Model

■ The standard VG distribution:

fV G(x; σ = 1, ν = 1, θ = 0, µ = 0) = 1 √ 2 exp(− √ 2|x|)

−6 −4 −2 2 4 6 0.2 0.4 0.6 0.8 x f(x) VG and standard Normal densities and log densities −6 −4 −2 2 4 6 −30 −25 −20 −15 −10 −5 x log (f(x)) standard VG standard Normal standard VG standard Normal

slide-32
SLIDE 32

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 32/35

LCDX VG Base Correlation

■ VG LCDX base correlation:

0.05 0.08 0.12 0.15 50 55 60 65 70 75 80 Tranch base correlation (%) Base Correlation : Lévy vs Gaussian LCDX.NA.9−V1 29Feb08 Gaussian Lévy VG 0.05 0.08 0.12 0.15 55 60 65 70 75 80 85 90 95 Tranch base correlation (%) Base Correlation : Lévy vs Gaussian LCDX.NA.9−V1 31Mar08 Gaussian Lévy VG

slide-33
SLIDE 33

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 33/35

LCDX Stable Model

■ The maximal skewed alpha stable distribution:

E[exp(iuX)] = exp(−(iu)α sec(πα/2)), 1 < α ≤ 2

−6 −5 −4 −3 −2 −1 1 2 0.1 0.2 0.3 0.4 x f(x) max skewed alpha stable and standard Normal densities and log densities −6 −5 −4 −3 −2 −1 1 2 −20 −15 −10 −5 x log f(x) max skewed alpha stable standard Normal max skewed alpha stable standard Normal

slide-34
SLIDE 34

Outline Introduction ABS Modeling LCDX Modeling

  • LCDS
  • Implied Prepayment and

Default Prob.

  • LCDX Modeling
  • LCDX Tranche Pricing
  • Conditional Probabilities
  • Recursive loss-prepayment

formula

  • LCDX Base Correlation
  • LCDX VG Model
  • LCDX VG Base Correlation
  • LCDX Stable Model
  • LCDX Stable Base

Correlation Conclusion Wim Schoutens, 09-09-2008 Linz, Austria - p. 34/35

LCDX Stable Base Correlation

■ Finite Moment log stable distribution (α = 1.5) LCDX base correlation:

0.05 0.08 0.12 0.15 50 55 60 65 70 75 80 Base "Correlation": Gaussian vs Max Skewed Alpha Stable LCDX.NA.9−V1 29Feb08 Gaussian Max Skewed Alpha Stable 0.05 0.08 0.12 0.15 55 60 65 70 75 80 85 90 95 Base "Correlation": Gaussian vs Max Skewed Alpha Stable LCDX.NA.9−V1 31Mar08 Gaussian Max Skewed Alpha Stable

slide-35
SLIDE 35

Outline Introduction ABS Modeling LCDX Modeling Conclusion

  • Conclusion

Wim Schoutens, 09-09-2008 Linz, Austria - p. 35/35

Conclusion

■ Lévy based models are introduced for modeling , ABSs, CDOs and LCDX. ■ Lévy base correlation is much flatter. ■ Lévy models are capable of describing in a more realistic way the risks

present. CONTACT: Wim Schoutens K.U.Leuven Celestijnenlaan 200 B B-3001 Leuven, Belgium Email: wim@schoutens.be Technical reports and more info on: www.schoutens.be