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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
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
Overview
1 Motivation
2 The basic model
3 Extension to a dynamic frailty model
4 Conclusion
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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).)
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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?
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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).
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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 .
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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%.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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)
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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).
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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”.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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%.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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. ↘).
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
Performances in-sample
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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)
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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).
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
”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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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)
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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).
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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 ).
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
⇒ 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
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic model
Extension to a dynamic frailty modelConclusionReferences
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.
Sbai (joint work with Delloye and Fermanian) Dynamic Frailties and Credit Portfolio Modeling
MotivationThe basic modelExtension to a dynamic frailty modelConclusionReferences