Dynamic Frailties and Credit Portfolio Modelingcermics.enpc.fr/~sbai/Papiers/Dallas2007.pdf · Default is the ﬁrst jump of an exogenous point process with ... dependence between

MotivationThe basic model

Extension to a dynamic frailty modelConclusionReferences

Dynamic Frailties and Credit Portfolio Modeling

M. Sbai1

Joint work with M. Delloye2 and J-D. Fermanian3

1IXIS CIB now CERMICS-ENPC2IXIS CIB now DEXIA

3IXIS CIB now BNP Paribas

Decision and Risk Analysis Conference21-22 May 2007

Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling



Overview

1 Motivation

2 The basic model

3 Extension to a dynamic frailty model

4 Conclusion




The use of Credit Portfolio Models in banks

Credit Portfolio Models are key-tools for(i) Active portfolio management(ii) Economic capital calculations (Var, EVar . . . )(iii) Pricing complex credit derivatives (CDOs, nth-To-Default

. . . )We are concerned here by the ”pure” risk measurement point ofview (ii).⇒ We will work exclusively under the historical probability.




Commercial softwares

For several years, some models have emerged in the financialcommunity. Among them,

Creditmetrics (JP Morgan),Credit Portfolio manager (KMV),CreditRisk+ (Credit Suisse Financial Products),CreditPortfolioView (McKinsey).Portfolio Risk Tracker (S&P)

(See Crouhy et al. (200) or Koyluoglu and Hickman (1998).)




Commercial softwares

Such models belong to the same class of factor models (Freyand McNeil [1]): the dependence between defaults is due tosome underlying common random variables (the factors).

The first three ones share some similarities and can bemapped to each other (Gordy [2]).

Such models have inspired Basel 2 proposals (theone-factor-”Basel model”).Open question : Which benchmark for Basel 3?




Modelling default I

Structural-type models (Merton, 1974)

Debt

Default time Time

Ass

et V

alue

Default : the firm’s asset value falls below the debt level.




Structural-type models (Merton, 1974)

Dependence between default events is due to the dependenceof the firms asset value processes (factor models).

Pros and Cos+ A widely known theoretical framework.+ An intuitive measure of dependence (correlation

coefficients).

- The asset value of a firm is not observable.- Common use of equity returns as proxies for asset returns.- Specification of the factors and calibration issues.




Modelling default II

Intensity-based models (Duffie and Singleton, 1999)

Also termed reduced-form or hazard rate models.Idea : directly model the default itself.Default is the first jump of an exogenous point process with

a random intensity λt = limdt→0

P (t ≤ τ ≤ t + dt |τ ≥ t)dt

.

Dependence between default events is due to thedependence between default intensities of the firms(λit = λ(t , Xt , U

it ) with Xt a vector of common covariates).




Intensity-based models (Duffie and Singleton, 1999)

Pros and Cos+ Some (macro or micro)-economic variables explain the

default events.+ A straight description of the default times law, without any

assumption on the firm behaviors (a priori a lighter modelrisk).

+ Calibration with respect to observable data.

- The generation of high dependence levels is recognized asa difficult task (Schönbucher, 2003).

- Necessity to exhibit ”good” explanatory variables.




Why do we choose an intensity-based model?

All the previous models generate the same patterns of lossdistributions (Koyluoglu and Hickman (1998), Gordy (2000). . . )

⇒ the key-point is calibration.

Moreover, in structural modelsThe correlation levels obtained empirically by calibrating onrating transitions data only are often far below thoseobtained with equity return indices (Hamerle et al. (2003)).Strong sensitivity of correlation estimates w.r.t firm sizes,portfolio heterogeneity and time horizon.Equity returns can be misleading because the include riskpremiums.




Specification of the model

A reduced-form model in the sense of Duffie andSingleton (1999), as a particular case of the Cox model(Lando (1998), Lando and Skoderberg (2002)).We modelize simultaneously every rating transition (Acompeting risks model) : at every time t , any firm i isfaced with 7 underlying risks (risks of rating changes,including default).Assumption : the associated time durations areindependent conditionally on the explanatory variables.




Specification of the model

λhji(t) : the intensity for the transition from h to j and for thefirm i .As in Kavvathas (2000), for every firm i , every time t andevery couple of transitions (h, j), h 6= j ,

λhji(t |z) = λhj0(t) exp(β′hjzhji(t)

)where

λhj0 is an unknown deterministic function (baseline hazardfunction, assumed constant).βhji is a parameter vector.zhji(t) is a vector of (common and idiosyncratic) factors attime t .




Building a vector of common factors

Intuitively, it is well understood that default probabilitiesdepend on the overall economic situation.See Keenan and al. (1999), Helwege and Kleiman (1996),and more generally the annual reports from Standard andPoors or Moody’s.More generally, rating transitions (including default) areinfluenced by some macroeconomic variables : see Nickelet al. (1998), Kim (2002), Bangia and al. (2000). . .

⇒ A challenging task is to identify the main explanatoryvariables that drive rating transitions and defaults.




Building a vector of common factors

We focus on transitions to defaults (crude but convenientassumption) : see Couderc and Renault (2005).

A simple linear regression to explain the monthly default ratesof US speculative firms (1986-2002) brought out four variables :

The annual variation rate of the index of industrialproduction in the USA,the annual variation rate of the S&P’s 500 index,the difference between the short (3 months) and the long(10 years) US government rates (the ”slope” of the IRC),the 3-month short US government rate.

These variables has been lagged.

The in sample fit is excellent : R2 = 87%.




0

1

2

3

4

5

6

86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 020

1

2

3

4

5

6 Actual Fitted

Monthly default rates (US speculative grades firms)




Building a vector of idiosyncratic variables

Additional firm-specific variables have been added as dummies:

the internal ratingto be a firm that is headquartered outside West Europe,USA or Canada,to be a financial firm,

and, following Lando and Skoderberg (2002),to have been upgraded/downgraded for less than one year(which try to capture non-Markovian features).




Parameters estimation

Full maximum likelihood estimation is possible (seeAndersen et al. (1993)) : L =

∏ni=1 Li , with

Nhji(t) : the number of transitions from h to j for the firm ibetween 0 and t .Yhi(t) : an indicator variable valuing 1 when i has rating h attime t−, and 0 otherwise.

Li =

∏t

∏j 6=h

(λhji(t |z)

)dNhji (t) exp−∑

(h,j)|j 6=h

∫ ∞0

Yhi(u)λhji(u|z) du

Right-censoring process : end of the observation period orentry into the category ”Not Rated”.




Assumption : The censoring variables have beenindependent from the underlying risks.The log-likelihood can be split into a sum over every (h, j)⇒ we can estimate the parameters separately.For example, if βhj = 0 then

λ̂hj0 =

∑ni=1

∑t dNhji(t)∑n

i=1∫∞

0 Yhi(u) du=

Number of transitions from h to jOccupation time of the rating state h

The CreditPro historical database (S&P):More than 10000 firms (60% USA)More than 15000 rating transitionsHistorical transitions from 1981




Results

Concerning the transition from rating CCC to default :

λCCC→D(t) = exp(−2.4− 4.7 ∗ IPIt− 0.41 ∗ S&Pt − 0.2 ∗ slopet− 0.17 ∗ short ratet − 0.62 ∗ 1neither US, EU− 0.12 ∗ 1bank + 1.3 ∗ 1 downgraded).

When the ”variation rate of IPI” ↗ 1% from the previous month,the default intensity for a CCC firm ↘ 4.7%.

When the short rate increases from 3% to 4%, the defaultintensity is down 17%.




More generallyThe IPI or the S&P index ↗⇒ the probability ofdowngrades (including default) ↘.The explanatory power of our variables is stronger fordowngrades than for upgrades.The transitions from AAA towards AA and conversely areparticular (”boosting” effect of the IPI rate).When the interest rate curve ↗, there are fewer defaultsand more rating transitions.Banks are more easily upgraded than the others, and lessstrongly downgraded.When a firm has been upgraded (resp. downgraded) oneyear before, the probability for subsequent upgrades ↘(resp. ↘).




The computation of transition matrices

Start : Intensity matrices Ii(t) = [λhji(t)]1≤h,j≤p.By definition, λhhi = −

∑j 6=h λhji(t).

⇒ Estimator for monthly transition matrices at t for firm i :

P̂i(t , t + 1) = Idp + Îi(t).

so, for arbitrary dates t1 and t2 (in months),

P̂i(t1, t2) =t2−1∏k=t1

P̂i(k , k + 1).

An annual transition matrix is get by the composition of 12successive monthly transition matrices.




Performances in-sample




Performances in-sample

in % AAA AA A BBB BB B CCC DAAA 91.8 7.39 0.68 0.11 0.06 0.00 0.00 0.00

AA 0.61 90.7 7.91 0.58 0.08 0.05 0.01 0.00A 0.06 1.97 91.7 5.61 0.45 0.19 0.02 0.01

BBB 0.03 0.23 3.85 89.9 4.99 0.85 0.09 0.08BB 0.03 0.09 0.48 5.13 83.8 8.58 1.29 0.58

B 0.00 0.06 0.19 0.46 5.01 83.1 7.28 3.93CCC 0.04 0.01 0.22 0.31 0.70 6.28 57.8 34.63

Yearly transition matrix 1981-2004 as provided by our model

in % AAA AA A BBB BB B CCC DAAA 91.8 7.50 0.48 0.12 0.06 0.00 0.00 0.00

AA 0.65 90.2 8.30 0.62 0.05 0.12 0.02 0.01A 0.05 2.20 91.0 5.98 0.46 0.18 0.04 0.05

BBB 0.03 0.24 4.26 89.0 5.01 0.87 0.22 0.33BB 0.03 0.09 0.39 5.91 82.8 8.26 1.12 1.36

B 0.00 0.08 0.24 0.33 5.67 82.1 4.97 6.63CCC 0.10 0.00 0.30 0.50 1.59 10.43 53.0 34.06Yearly transition matrix 1981-2004 as provided by S&P (CreditPro 7.0)




in % 1 year 2 years 5 yearsIG SG crossed IG SG crossed IG SG crossed

Model 0.34 1.22 0.04 0.36 2.11 0.21 0.39 1.64 0.24S&P 0.19 1.43 0.36 0.35 2.32 0.64 0.33 2.55 0.67

Correlations as provided by our model and by S&P (non financial US-UE firms)

Main criticsTail of the loss distribution is not sufficiently fat comparedwith standards.Correlation coefficients are not high enough.Difficulty with the particular transition BBB → B.




Surely, some relevant explanatory variables have not beentaken into account :

systemic : inflation, unemployment . . .idiosyncratic : quality of the management, financial ratios. . .

⇒ Extension to frailty models.See Clayton and Cuzick (1986), Hougaard (2000) . . . , instatistics.In finance : Métayer (2004), Schönbucher (2005), Fermanianand Sbai (2005) and recently Duffie et al. (2006).




”Frailty” : Latent variable that act multiplicatively on the intensityof default.The intensity becomes : λhji(t |z) = γhji(t)λhj0(t) exp(β′hjZhji(t)).The frailty variables will be dynamic ⇒ a frailty process ratherthan static frailties.Indeed : as the observed macro factors Z , the unobservedfactors that drive the credit risk should be time-dependentMoreover, empirical results show a ”law of large numbers”effect (compensation between periods) which prevents thegeneration of high dependence levels.




Estimation

The ”complete” likelihood is Lc =∏n

i=1 Lci with

Lci =

∏t

∏j 6=h

λhji(t |Z )dNhji (t) exp

−∑j 6=h

∫ ∞0

Yhi(u)λhji(u|Z )du

To simplify : γt ≡ γhjit

L = Eγ(Lc)

= C(β)Eγ

(∏T0t=1 γ

Pni=1

Pj 6=h ∆Nhji (t)

t

· e−γtPn

i=1P

j 6=hR t

t−1 Yhi (u)eβhj0+β

Thji Xhji (u)du

),

whereC(β) =∏n

i=1∏

t∏

j 6=h e(βhj0+β

Thji Xhji (t))dNhji (t)




Specification of the frailty dynamics

The likelihood cannot be simplified : no closed-form formulas,except in the special case of constant frailties.The dynamic frailty model :

γhij1 = γ̃hij1, γhijt = γhij,t−1 · γ̃hij,twhere the γ̃hijt are drawn independently for every t :

γ̃hij,t ∼ G(α, α),E[γ̃hij,t ] = 1, Var(γ̃hij,t) = 1/α.

Ref: Yue and Chan (1997), Paik et al. (1994) or Yau andMcGilchrist (1998).Difficulty with such models : the inference (the lack of tractableformulas).p = density of the vector of frailties

γT0 =

γ1...γT0

.Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling



p(dγT0) =T0∏

t=1

g(

α,α

γt−1

)(γt) dγ1 · · ·dγT0 ,

where g(α, β) = density of a gamma r.v. G(α, β)g(α, β)(x) = β

αxα−1Γ(α) e

−βx1R+(x).

⇒ L =∫R

T0L(θ) dγ1 · · ·dγT0 ,

where

L(θ) = C(β)T0∏

t=1

γPn

i=1P

j 6=h ∆Nhji (t)t

· e−γtPn

i=1P

j 6=hR t

t−1 Yhi (u)eβhj0+β

Thji Xhji (u)du g

(α,

α

γt−1

)(γt).




Maximization of L: The EM algorithm, based on Monte CarloMarkov Chains simulation techniques.Starting from an “arbitrary” θ0, and assuming we have foundθ1, . . . , θk , we need to maximize the Q(·|θk ) criterion

Q(θ|θk ) =∫R

T0ln(L(θ))p(dγT0 |Y , θk ), (1)

where p(·|Y , θk ) = density of the frailties vector γT0 knowing allthe observations Y (all the rating transitions) and assuming thevalue of our parameter is θk .At each step, we approximate the integral in equation (1) by asum, by a usual Monte Carlo procedure. Therefore we need todraw in the conditional law p(·|Y , θk ).




⇒ An Hastings-Metropolis algorithm with random walkKnowing γT0s ,

1 Generate ys ∼ q(· − γT0s ).2 Generate

γT0s+1 =

{ys with probability ρ = min (1,

f (ys)

f (γT0s )

),

γT0s with probability 1− ρ.

q ∼ unif [−0.1, 0.1]T0 .⇒ we simulate a Markov chain (γT01≤t≤S) whose stationary law isf = p(·|Y , θk ).By doing the same procedure S times, we approximate thecriterion Q(θ|θk ) by

Q̃(θ|θk ) =1S

S∑s=1

ln(L(θ))(γT0s ),

that we maximize with respect to the parameters θ to get θk+1.Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling



Note that the previous quantity ρ can be written relatively simplybecause

p(γT01 |Y , θ)p(γT02 |Y , θ)

=

T0∏t=1

(γ1,tγ2,t

)ν(t)+α−1·

(γ2,t−1γ1,t−1

)αe

b(t)(γ2,t−γ1,t )−α(γ1,t

γ1,t−1−

γ2,tγ2,t−1

)

where{ν(t) =

∑ni=1

∑j 6=h ∆Nhji(t),

b(t) =∑n

i=1∑

j 6=h∫ t

t−1 Yhi(u)eβ̄hj0+β̄

Thji Xhji (u)du.

Only two groups: one notch downgrades and one notchupgrades. The estimated α is 23.6 (downgrades) and 52.0(upgrades).In the long run, the process γt can reach relatively high values.Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling



Statistical indicators of the dynamic frailty processes :simulation of 1000 trajectories, with α = 100.

Mean St.dev. Quantile 95% Maximum1 year 0,999 0,101 1,171 1,431

2 years 0,999 0,142 1,248 1,6943 years 1,000 0,175 1,310 1,9874 years 1,003 0,203 1,370 1,9585 years 1,003 0,229 1,415 2,318

10 years 1,008 0,332 1,615 3,22515 years 1,003 0,402 1,768 4,96220 years 1,001 0,464 1,873 5,67025 years 1,001 0,526 2,008 6,15330 years 1,003 0,581 2,121 7,572




Illustration : a simple portfolio (50 firms, different ratings from AAA to CCC), with longterm 30-years constant equal exposures.

Loss distributions given by the basic model and the frailty model.




1 A few number of assumptions2 They allow the data “explaining the reality” by themselves3 Good empirical results.

Weaknesses of such reduced-form models :

1 In terms of forecasts : a sensitivity with respects to thecovariate process

2 A relatively difficult estimation procedure.




References I

R. Frey and A. McNeil.Modelling dependent defaults.ETH E-Collection. 2001.

M. Gordy.A comparative anatomy of credit risk models.Journal of Banking and Finance, 24:119–149, 2000.


MotivationThe basic modelExtension to a dynamic frailty modelConclusionReferences

Documents

Dynamic Frailties and Credit Portfolio Modelingcermics.enpc.fr/~sbai/Papiers/Dallas2007.pdf · Default is the ﬁrst jump of an exogenous point process with ... dependence between