Self-Exciting Corporate Defaults: Contagion or Frailty?

Transcription

1 1 Self-Exciting Corporate Defaults: Contagion or Frailty? Kay Giesecke CreditLab Stanford University giesecke Joint work with Shahriar Azizpour, Credit Suisse

2 Self-Exciting Corporate Defaults 2 Defaults cluster 2 Number of Defaults (Moodys Senior Rated) Years(197 26)

3 Self-Exciting Corporate Defaults 3 Sources of clustering Firms exposure to observable factors: doubly-stochastic models Defaults are conditionally independent Das, Duffie, Kapadia & Saita (27): not enough Firms exposure to unobservable factors: frailty models Bayesian updating of frailty distribution at events Duffie, Eckner, Horel & Saita (27): increases clustering Firms s exposure to default events: contagion models Failure of a firm tends to weaken the others Complex web of contractual relationships in the economy

4 Self-Exciting Corporate Defaults 4 Feedback from events

5 Self-Exciting Corporate Defaults 5 Contagion or frailty? Both generate similar statistical effects in conditional default rates Jumps at events Yet they have distinct economic foundations Contagion: contractual linkages among firms Frailty: asymmetric information This paper: Develop, estimate and test a model of correlated event timing that incorporates contagion and frailty Understand the relative empirical importance for U.S. corporate defaults of these phenomena

6 Self-Exciting Corporate Defaults 6 Preview of empirical results A default is estimated to have a significant influence on the conditional default rates of surviving firms Rejection of doubly-stochastic hypothesis Contagion and frailty are roughly equally important sources for the feedback from events Can explain the dramatic time-variation of U.S. corporate default rates during

7 Self-Exciting Corporate Defaults 7 Default data: 1374 events on 99 dates

8 Self-Exciting Corporate Defaults 8 Events per default date

9 Self-Exciting Corporate Defaults 9 Self-exciting model of event timing (Ω, F, P ) a complete probability space and F = (F t ) t a complete information filtration satisfying the usual conditions Events arrive at ordered stopping times T n that generate a non-explosive counting process N with F-intensity λ N λ sds is an F-local martingale λ evolves through time according to the self-exciting model dλ t = κ(c λ t )dt + σ λ t dw t + δdl t where W is an F-standard Brownian motion and N L = l(d n ) n= where l is a positive and bounded weight function and D n F Tn is the number of defaults at T n

10 Self-Exciting Corporate Defaults 1 Sample path of (λ, L) Intensity Loss Time

11 Self-Exciting Corporate Defaults 11 Default correlation channels 1. Impact of an event on the surviving firms through δl 2. Exposure to a Feller diffusion risk factor W 3. Uncertainty about the current value of W This occurs when W is not adapted to the observation filtration G = (G t ) t, which may be much coarser than F In this case W is a frailty whose values must be filtered from the information in G

12 Self-Exciting Corporate Defaults 12 Filtered intensity The econometrician s filtered intensity h of N is a G-adapted process h such that N h sds is a G-local martingale It is given by the optional projection of the complete information intensity λ onto G, assuming G is fine enough to distinguish N h t = E[λ t G t ], almost surely The filtered intensity h is revised at events Contagious impact of the event (inherited from λ) Bayesian updating of the G-conditional distribution of W

13 Self-Exciting Corporate Defaults 13 Filtered likelihood G is generated by the (T n, D n ) and a covariate process X The parameter vector to be estimated is (θ, γ, ν), where θ = (κ, c, σ, δ, λ, w) represents the intensity parameters For the sample period [, τ], the likelihood function is f τ (N τ, T ; θ D, X) g(d; γ) p τ (X; ν) f τ (, ; θ D, X) is the conditional density of N τ and T = (T 1,..., T Nτ ) given D = (D 1,..., D Nτ ) and the covariate path over [, τ] g( ; γ) is the probability function of D p τ ( ; ν) is the density of the covariate path over [, τ] The three terms can be maximized separately to give the full likelihood estimates

14 Self-Exciting Corporate Defaults 14 Filtered likelihood To evaluate the density f τ (N τ, T ; θ D, X), we transform the point process (L, F) into a standard F-compound Poisson process by an equivalent change of measure With the standard abuse of notation, f τ (N τ, T ; θ D, X) = Ê[ Z 1 τ e τ N τ, T, D, X ] where Ê denotes expectation with respect to the measure P on F τ defined by the density Z τ, where ( t t ) Z t = exp log(λ s )dn s (1 λ s )ds Z τ is a function of (T n, D n ) n=1,...,nτ and {W t : t τ} If W can be identified from X or σ =, then f τ (N τ, T ; θ D, X) = Zτ 1 e τ

15 Self-Exciting Corporate Defaults 15 Filtered likelihood If W is a frailty that cannot be identified from X, then the conditional expectation is a nontrivial filter Since W is a ( P, F)-standard Brownian motion that is P -independent of L, we can show that Ê [ Zτ 1 e τ Nτ, T, D, X ] [ N τ = Ê k=1 λ Tk φ Tk 1,T k (λ Tk 1, λ Tk )φ Tn,τ (λ Tn, λ τ ) Here, for constants a b τ and positive v and w, [ ( φ a,b (v, w) = Ê exp b a ] λ s ds) λ a = v, λ b = w ] N τ, T, D which can be expressed explicitly (Broadie & Kaya (26)) since λ follows an F-Feller diffusion between events

16 Self-Exciting Corporate Defaults 16 Goodness-of-fit tests via time change We wish to assess the goodness-of-fit of a specification (N, λ, G) Meyer s (1971) theorem implies that the G-counting process (N, h) can be transformed into a standard Poisson process by a change of time that is given by the G-compensator A = h sds If λ and G are correctly specified, then the (A Tn ) form a standard Poisson process in the time-changed filtration generated by (A 1 t ) We test the Poisson property using two tests Kolmogorov-Smirnov test Prahl s (1999) test The tests are applied in-sample and out-of-sample

17 Self-Exciting Corporate Defaults 17 Goodness-of-fit tests via time change Time S 6 (ω) S 5 (ω) S 4 (ω) S 3 (ω) S 2 (ω) S 1 (ω) Compensator T 1 (ω) T 2 (ω) T 3 (ω) T 4 (ω) T 5 (ω) T 6 (ω)

18 Self-Exciting Corporate Defaults 18 Zero-factor model: σ = λ is G-adapted so h = λ and likelihood is in closed form We solve sup θ log f τ (N τ, T ; θ D, X) by grid search over discretized parameter space Quadratic weight function l(n) = n + wn 2 fits best MLEs: ˆκ = 1.84, ĉ = ˆλ = 5.48, ˆδ =.43, ŵ =.45 Observations An event has a significant impact on fitted default rates The fitted λ responds quickly to event bursts The simple 4-parameter model captures the substantial time-series variation of default rates during

19 Self-Exciting Corporate Defaults 19 Zero-factor model: σ = Fitted intensity λ vs. events per year

20 Self-Exciting Corporate Defaults 2 Zero-factor model: σ = Empirical distribution of re-scaled inter-event times

21 Self-Exciting Corporate Defaults 21 Zero-factor model: σ = QQ plot of the re-scaled inter-event times vs. standard exponential

22 Self-Exciting Corporate Defaults 22 Zero-factor model: σ = 1Y Forecast conditional portfolio loss distribution (out-of-sample)

23 Self-Exciting Corporate Defaults 23 Zero-factor model: σ = 1Y Forecast conditional portfolio loss distribution vs. actual events 4 35 (1%,99%)quantile (5%,95%)quantile Mean Observed

24 Self-Exciting Corporate Defaults 24 Zero-factor model: σ = Forecast portfolio value at risk (out-of-sample)

25 Self-Exciting Corporate Defaults 25 Zero-factor model: σ = G τ -conditional portfolio loss surface (LGD uniform on {.4,.6,.8, 1}).6.5 Probability Horizon Loss 1 15

26 Self-Exciting Corporate Defaults 26 One-factor model: σ > λ is not always G-adapted Non-informative covariate X: W is independent of X, and therefore not G-adapted (1) Contagion, (2) Factor exposure to W, (3) Frailty Informative covariate X: W is G-adapted since it can be recovered from X (1) Contagion, (2) Factor exposure We treat these cases separately to understand the relative empirical importance of contagion and frailty

27 Self-Exciting Corporate Defaults 27 One-factor model with non-informative X λ is not G-adapted and the likelihood must be filtered MLEs: ˆκ = 1., ĉ = ˆλ = 6.2, ˆσ = 3.5, ˆδ =.2, ŵ =.5 Compare with zero-factor model estimate ˆδ =.43 The filtered intensity h t = E [ λ t G t ] = Ê [ Z 1 t λ t G t ] Ê [ Z 1 t G t ], t τ Jumps at T n due to contagion and Bayesian updating of the G-conditional distribution of W Deterministic between events

28 Self-Exciting Corporate Defaults 28 One-factor model with non-informative X Filtered intensity h t = E[λ t G t ] vs. zero-factor intensity

29 Self-Exciting Corporate Defaults 29 One-factor model with non-informative X Smoothed intensity H t = E[λ t G τ ] vs. zero-factor intensity

30 Self-Exciting Corporate Defaults 3 One-factor model with non-informative X Empirical distribution of re-scaled inter-event times: fit deteriorated

31 Self-Exciting Corporate Defaults 31 One-factor model with informative X λ is G-adapted so h = λ (no frailty) We explore two covariates: S&P 5 index value, 1Y Treasury yield Modeled as G-Feller diffusions driven by W Can recover W from X using covariate MLE Treating the estimated W as though error-free, we then estimate λ as in the complete information case MLEs are similar to that of zero-factor model S&P 5 covariate performs slightly better than yield

32 Self-Exciting Corporate Defaults 32 One-factor model with informative X Fitted intensity λ for two covariate choices Default Dates Hawkes Yield S&P

33 Self-Exciting Corporate Defaults 33 Discussion and conclusion An event is estimated to have a significant impact on fitted U.S. default rates, in all model variants Implications for modeling of correlated default risk We found that contagion and frailty are roughly equally important sources for this impact Feedback through contagion or frailty is necessary to fit the dramatic time variation of U.S. default rates during , in-sample and out-of-sample Feedback explains the excess event clustering found by Das, Duffie, Kapadia and Saita (27) for doubly-stochastic models

34 Self-Exciting Corporate Defaults 34 Discussion and conclusion The simple zero-factor model (σ = ) is hard to beat The one-factor model without frailty does about as well Indicates the information content of event times Do we need frailty in our self-exciting intensity model λ? Fit is worse, the estimation is challenging No new statistical features relative to complete information self-exciting model λ Re-interpret our self-exciting (λ, F) as filtered intensity in a frailty model in a super-filtration H F F then takes the role of the observation filtration Feedback jumps of λ can be interpreted in terms of contagion, or Bayesian updating of the frailty distributions

35 Self-Exciting Corporate Defaults 35 References Broadie, Mark & Ozgur Kaya (26), Exact simulation of stochastic volatility and other affine jump diffusion processes, Operations Research 54(2), Das, Sanjiv, Darrell Duffie, Nikunj Kapadia & Leandro Saita (27), Common failings: How corporate defaults are correlated, Journal of Finance 62, Duffie, Darrell, Andreas Eckner, Guillaume Horel & Leandro Saita (27), Frailty correlated default. Working Paper, Stanford University. Meyer, Paul-André (1971), Démonstration simplifée d un théorème de Knight, in Séminaire de Probabilités V, Lecture Note in Mathematics 191, Springer-Verlag Berlin, pp Prahl, Jürgen (1999), A fast unbinned test of event clustering in

36 Self-Exciting Corporate Defaults 36 poisson processes. Working Paper, Universität Hamburg.