SLIDE 1 Large portfolio losses; A dynamic contagion model.
Marco Tolotti Universit` a Bocconi, Milano Scuola Normale Superiore, Pisa based on joint work with:
- P. Dai Pra, W. Runggaldier and E. Sartori
(Universit` a di Padova) Stochastic processes: theory and applications. In honor of the 65th birthday of Wolfgang J. Runggaldier, Bressanone July 18, 2007.
SLIDE 2 Outline
- The contagion process in Finance and Economics:
examples from different disciplines (overview);
- Motivations. Credit risk management issues:
Large portfolio losses analysis and credit crises;
- Modeling aspects: Interacting intensities, Markov processes;
- Main results: Asymptotic results, Equilibria, Fluctuations;
- Back to Finance: An example and numerical simulations.
SLIDE 3 What do we mean by contagion? Contagion: “Spread of an activity or a mood through a group.” (From the Penguin Dictionary of Psychology). Quoting Allen and Gale (01): “Small shocks, which initially affect
- nly a few institutions, spread by contagion to the rest of the
financial sector and then infect the large economy” → Propagation of financial distress from one agent (institution, firm, obligor, ...) to the others through direct (or informational) relationships.
SLIDE 4
... the contagion process (cont’d) Effects: Systemic risk, Bank runs, Domino effects, Credit crises. Or in the social sciences literature: Herding, Peer pressure, ... The goal: (see e.g. Durlauf 01 on complexity in Economics): Try to explain macroeconomic (aggregate) emergent phenomena (as path dependence, non ergodicity, power laws, persistency, heavy tails) modeling interdependencies across individuals at the micro-structure level.
SLIDE 5 Some examples (from the literature) 1) Financial Contagion, Allen and Gale (2001) Liquidity shocks: When liquidity demand is higher than what expected, the lack of liquidity may develop throughout the whole banking system due to interregional cross holdings. (systemic risk) 2) Random Economies with interacting agents, F¨
F¨
- llmer & Schweizer (1993), ...
The preferences of an individual depend on the preference of his peers, randomness in individual preferences may affect the aggregate: → Convergence to a fundamental value on the long-run, fluctuations on the short (explain bubbles?)
SLIDE 6 Some examples (from the literature) cont’d. 3) Herding behavior and aggregate fluctuations in financial markets, Cont and Bouchaud (2000) Communication structure (e.g. imitation, herding behavior) btw agents gives rise to large fluctuations in the aggregate excess demand (heavy tails, excess kurtosis, power laws) 4) Discrete choice models, Brock and Durlauf (2001) The aim is to embody social interaction in the decision process. The marginal utility of agent i explicitly depends on the actions of
- ther agents. In particular:
U(ωi) = u(ωi) + S(ωi, ¯ mi) + ǫ(ωi). where ¯ mi is the subjective expected value of the choices of the
→ Under some conditions multiple equilibria are found.
SLIDE 7
Contagion in credit risk 1) Counterparty risk, Jarrow and Yu (2001) The default of a firm’s counterparty affects its own probability of default. λA
t := lim h→0
1 hP(τA ≤ t + h|τA > t) = b1 + b21{t≥τB}. 2) Interacting default intensities, Frey and Backhaus (04-06) Default intensity depends on the state of the system Y and on some macro-factors Ψ. Hence λA
t = 1{t≤τA}hA(Yt, Ψt).
3) Contagion and aggregate losses, Giesecke & Weber (04-05) Liquidity of firm A depends on the liquidity of a group of neighbors. → Study large portfolio losses (i.e., let N be ”large”).
SLIDE 8
Modeling context Interacting intensities: Transition rates of a single firm depend on the current state of the other obligors. → Contrary to models relying only on macroeconomic (exogenous) factors we have direct interaction: (no conditional independence of the events of default!).
Jarrow & Yu: “A default intensity that depends linearly on macroeconomic variables is unlikely to account for clustering of defaults”.(JOF 01)
SLIDE 9
Modeling context Interacting intensities: Transition rates of a single firm depend on the current state of the other obligors. → Contrary to models relying only on macroeconomic (exogenous) factors we have direct interaction: (no conditional independence of the events of default!).
Jarrow & Yu: “A default intensity that depends linearly on macroeconomic variables is unlikely to account for clustering of defaults”.(JOF 01)
→ Macro indicators of the general “health” of the system can here be inferred (endogenously) from the microeconomic model.
SLIDE 10 Our goals
- 1. The credit quality of a firm is directly influenced by the state of
- ther firms (contagion).
→ Obtain a dynamic description of the contagion process. → Describe a mathematical framework able to explain clustering of defaults or credit crises (modeling issues)
- 2. Investigate the losses that a bank may suffer in a large portfolio
due to deterioration of credit quality of the obligors. → Compute the distribution of losses and risk measures. (risk management issue)
SLIDE 11 The model N firms active on the market.
- 1. The credit quality itself of firm i is described by a binary variable
σi ∈ {−1; +1}. View it as a two rating classes model with σ a rating indicator (generalizations to more values for σ possible) → Low value of σ: higher probability of not being able to pay back
SLIDE 12 The model N firms active on the market.
- 1. The credit quality itself of firm i is described by a binary variable
σi ∈ {−1; +1}. View it as a two rating classes model with σ a rating indicator (generalizations to more values for σ possible) → Low value of σ: higher probability of not being able to pay back
- bligations.
- 2. Strength of firm i described by a binary variable ωi ∈ {−1; +1}.
View it as a financial distress indicator (buffer capacity).
(E.g. Sign of cash balances: Cetin et al. (2004))
→ Fundamental indicator: assumed not to be directly observable. → It introduces heterogeneity in the model
SLIDE 13 Time evolution and vehicle of contagion → To determine the time evolution on a generic interval [0, T] of the “state” of the system, i.e. (σi(t), ωi(t))i=1,··· ,N ∈ D2N[0, T] we need to specify the rates/intensities for the transitions σi → −σi, ωi → −ωi.
We let the interaction depend on the global health indicator (endogenous global factor) mσ
N(t) := 1
N
N
σi(t)
SLIDE 14 Time evolution and vehicle of contagion → To determine the time evolution on a generic interval [0, T] of the “state” of the system, i.e. (σi(t), ωi(t))i=1,··· ,N ∈ D2N[0, T] we need to specify the rates/intensities for the transitions σi → −σi, ωi → −ωi.
We let the interaction depend on the global health indicator (endogenous global factor) mσ
N(t) := 1
N
N
σi(t)
- The vehicle of contagion is given by
ωi
→ σi
rating class
→ mσ
N global health indic.
→ ωj
SLIDE 15 Transition intensities for the system σi → −σi with intensity λi := e−βσiωi , β > 0 ωj → −ωj with intensity µj := e−γωjmσ
N ,
γ > 0 → The parameters β, γ indicate the strength of the interaction. → The transition from σ = −1 to σ = +1 is higher where ω = +1 (strong firm, positive cash balances). The transition from ω = −1 to ω = +1 is higher when mσ
N is
higher. → May be generalized to letting the rates depend on exogenous macroeconomic factors. (business cycles)
SLIDE 16 → The state variables form a continuous-time Markov chain on the configuration space {−1, +1}2N with infinitesimal generator L acting on f : {−1, 1}2N → R as Lf(σ, ω) =
N
λi
+
N
µj
- f(σ, ωj) − f(σ, ω)
- where σi = (σ1, ..., σi−1, −σi, σi+1, ..., σN);
analogously for ωj.
SLIDE 17 General results Concern the dynamics of the system (σi(t), ωi(t)); i = 1, ..., N and
N(t) for large N, more precisely:
- A. look for the limit (N → ∞) dynamics of the system
→ characterize a suitable Law of Large Numbers;
- B. study the equilibria of the limiting dynamics
→ detect phase transition, multiple equilibria;
- C. describe finite volume approximations
→ characterize a suitable Central Limit Theorem.
SLIDE 18 A crucial object: The empirical measure ρN ∈ M1(D2[0, T]). ρN = 1 N
N
δ{σi[0,T],ωi[0,T]} for a generic function f of the trajectory one has 1 N
N
f(σi[0, T], ωi[0, T]) =
Remark: Derive the empirical means: if f ≡ Πt(σi[0, T]) then
fdρN = mσ
N(t)
if f ≡ Πt(ωi[0, T]) then
fdρN = mω
N(t)
if f ≡ Πt(σi[0, T] · ωi[0, T]) then
fdρN = mσω
N (t)
SLIDE 19
Theorem 1. Let (σ(t), ω(t)) be the Markov process corresponding to the generator Lf(σ, ω) and with initial distribution s.t. (σi(0), ωi(0)), i = 1, ..., N are i.i.d. with law λ, then i) there exists Q∗ ∈ M1(D2[0, T]) s.t. ρN → Q∗ a.s. in the weak topology; ii) let q∗
t ≡ ΠtQ∗ then q∗ t is the unique solution of
∂qt
∂t
= Lqt, t ∈ [0, T] (∗) q0 = λ → relying on Theo 1, one can easily describe the dynamics of the aggregate variables
mσ(t) :=
σdq∗
t
mω(t) :=
ωdq∗
t
mσω(t) :=
σωdq∗
t
SLIDE 20
- B. Equilibria of the limiting dynamics
Proposition 2. Equation (∗) can be reduced to determining a solution of ( ˙ mσ(t), ˙ mω(t)) = V (mσ(t), mω(t)) (∗∗) with V (x, y) = ( 2 sinh(β)y − 2 cosh(β)x, 2 sinh(γx) − 2y cosh(γx) ). → Equilibria of (∗) and their stability are determined via (∗∗).
SLIDE 21 Theorem 3. i) For γ < coth (β), Equation (∗∗) has (0, 0) as unique equilibrium
- solution. The equilibrium (0, 0) is linearly stable, i.e., DV (0, 0)
(the Jacobian matrix) has strictly negative eigenvalues. ii) For γ = coth (β), (∗∗) has (0, 0) as unique equilibrium solution but the linearized system has a neutral direction, i.e., DV (0, 0) has one zero eigenvalue. Nevertheless, (0, 0) is globally asymptotically stable, i.e., for ev- ery initial condition (mσ(0), mω(0)) we have lim
t→∞(mσ(t), mω(t)) = (0, 0).
SLIDE 22
iii) γ > coth (β). The point (0, 0) is an equilibrium but it is a saddle point for the linearized system, i.e., the matrix DV (0, 0) has two non-zero real eigenvalues of opposite sign. Moreover (∗∗) has two linearly stable solutions (mσ
∗, mω ∗ ) and (−mσ ∗, −mω ∗ ) where mσ ∗
is the unique strictly positive solution of x = tanh (β) tanh (γx) and where mω
∗ = coth (β)mσ ∗.
The phase space [−1, 1]2 is bipartitioned by a smooth curve Γ containing (0, 0) such that [−1, 1]2 \ Γ is the union of two open disjoint sets Γ+ and Γ−, moreover lim
t→∞(mσ(t), mω(t)) =
(mσ
∗, mω ∗ )
if (mσ(0), mω(0)) ∈ Γ+ (−mσ
∗, −mω ∗ )
if (mσ(0), mω(0)) ∈ Γ− (0, 0) if (mσ(0), mω(0)) ∈ Γ
SLIDE 23 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Domains of attraction when β=1and γ=2.3 mσ mω
Γ+ Γ−
(−mσ
* ,−mω * )
Γ
(mσ
* ,mω * )
SLIDE 24
Description of a credit crisis For critical values of the parameters (i.e., in periods of high interaction on the market): → For certain values of the initial conditions the system is driven towards the asymptotic unstable equilibrium state (0, 0). After a suitable time (depending on the parameters) the system is captured by an unstable direction and moves towards a stable equilibrium. Moreover, during this transition the volatility of the system in- creases sharply before decaying to stationary values. This phenomenon can be interpreted as a credit crisis.
SLIDE 25 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Phase diagram of a trajectory of (mσ,mω) mσ
Γ+ Γ−
mσ(0),mω(0) Phase diagram of (mσ,mω) (−mσ
* ,−mω * )
Γ
SLIDE 26
A general CLT is also proved for the flow of projections of empirical measures: √ N(ρN(t) − (qt))t∈[0,T]. Proposition 4. The sequence of global health indicators (mσ
N)N
√ N(mσ
N(t) − mσ(t)) → N(0, Σx(t))
where Σx(t) is the first entry of a (well defined) covariance matrix. In particular P(mσ
N(t) ≥ M) ≈ N
√ Nmσ(t) − √ NM
.
SLIDE 27 Portfolio losses A bank holds a portfolio of financial positions issued by the N firms.
- Marginal loss for the i − th position at time t:
L(σi(t), Ψ) ∈ R+ ; i = 1, ..., N where Ψ is an (exogenous) factor. → Homogeneity: given σi = x, x ∈ {−1; 1} and Ψ ∈ R, losses are i.i.d. (Giesecke & Weber 04,05)
- Aggregated (portfolio) losses:
LN(t) =
N
L(σi(t), Ψ)
- Large portfolio losses at time t: Take N big but finite.
SLIDE 28
Example (Bernoulli mixture models) Portfolio consisting of N positions of 1 unit due at time T. → Exposure at default = 1 (simplification: only PD matters!) L(σi(T), Ψ) =
1 with prob P(σi(T), Ψ) with prob 1 − P(σi(T), Ψ) Possible specification for P: P(σ; Ψ) = 1 − exp{−k1Ψ − k2(1 − σ) − k3} ; ki > 0 with Ψ ∼ Γ(α; κ) (similar to CreditRisk+)
SLIDE 29 Theorem 5. Let Ψ ∼ Γ(α; κ) then P
N µ(T; mσ
q , ψ) − L
√ N
q , ψ)
d
fΨ(ψ) where µ(T; mσ
q , Ψ) and V (T; mσ q , Ψ) are simple transformations of
mσ
q and Σx(t); and where fΨ(·) is the density function of Ψ.
→ VaR-like and other risk measures can be easily computed.
SLIDE 30 Theorem 5. Let Ψ ∼ Γ(α; κ) then P
N µ(T; mσ
q , ψ) − L
√ N
q , ψ)
d
fΨ(ψ) where µ(T; mσ
q , Ψ) and V (T; mσ q , Ψ) are simple transformations of
mσ
q and Σx(t); and where fΨ(·) is the density function of Ψ.
→ VaR-like and other risk measures can be easily computed. Remark: Respect to standard factor models (CreditRisk+): → We have introduced direct contagion and this has an impact on both the expected number of firms in financial distress and the distribution of the losses in a large portfolio. → We take into account the possibility (for certain values of the parameters and initial conditions) of having a credit crisis: in short time mσ(t) suddenly falls to a (lower) value −mσ
∗.
SLIDE 31
3000 3500 4000 4500 5000 0.2 0.4 0.6 0.8 1 Excess loss: x Excess probability: P(LN>x) Comparing excess losses varying the parameters β=1.5, γ=0.6, Ψ=4.5 β=1.5, γ=1.1, Ψ=4.5 β=1.5, γ=1.1, Ψ∼Γ(2.25,2)
SLIDE 32 2 4 6 8 10 12 −1 −0.8 −0.6 −0.4 −0.2 time mσ
t
Trajectory of mσ
t and V(t)
2 4 6 8 10 12 5 10 15 20 x 10
4
time V(t)
V(t) V(t), t=2 V(t), t=10
mσ
t
mσ
t , t=2
mσ
t , t=10
SLIDE 33 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Phase diagram of a trajectory of (mσ,mω) mσ
Γ+ Γ−
mσ(0),mω(0) Phase diagram of (mσ,mω) (−mσ
* ,−mω * )
Γ
SLIDE 34 Summarizing
- We have modeled contagion (propagation of financial distress)
in a network of firms.
- We have described the equilibria of the asymptotic (N → ∞)
system and their stability. For certain values of the parameters a credit crisis has been detected.
- We have characterized the loss distribution of a large portfolio
via a time dependent Gaussian approximation (closed formulae!).
- We have provided an example where both contagion and macroe-
conomic effects are taken into account: Generalization of stan- dard factor models.
SLIDE 35 Next steps and developments
- Build a more comprehensive model:
- Generalize the variable ω;
E.g.: dωt = ωt[α(mσ
N(t))dt + vt(mσ N(t))dBt] + dJt.
- Weak the mean-field assumption by assuming that the jump
rate of ωi depends on mσ
N,i = N j=1 J
i
N, j N
tion J(·, ·) : [0, 1]2 → R.
- Study the critical case for long time horizons.
If γ = coth(β) the CLT does not hold any more (Σt explodes!). The following convergence in distribution holds: N
1 4
N(
√ Nt) − mσ( √ Nt)
N→∞
− → Z where Z is Non-Gaussian and has heavy tails.
- Provide a Poissonian type limiting result for the portfolio losses.
SLIDE 36
Thank you very much for your attention!
SLIDE 37
A large deviation principle A sequence of measures pn satisfies a LDP with rate function I(·) : X → [0, +∞] if for large n pn ∼ e−n infQ I(Q) hence, in our case, ρN satisfies as LDP in the sense that P(ρN ∈ A) ∼ e−N infQ∈A I(Q) It can be shown that there exists a (unique) measure Q∗ such that I(Q∗) = 0. Thus the law of large numbers is basically proved.
SLIDE 38
Theorem 5. Let
xN(t) = √ N(mσ
N(t) − mσ(t))
yN(t) = √ N(mω
N(t) − mω(t))
zN(t) = √ N(mσω
N (t) − mσω(t))
then (xN(t), yN(t), zN(t)) N→∞ − → (x(t), y(t), z(t)) in the sense of weak convergence of stochastic processes, where (x(t), y(t), z(t)) is a cen- tered Gaussian process, unique solution of the linear SDE
dx(t) dy(t) dz(t)
= A(t)
x(t) y(t) z(t)
dt + D(t)
dB1(t) dB2(t) dB3(t)
where B1, B2, B3 are independent Brownian motions and A(t), D(t) are 3 × 3 matrices such that D(t)DT(t) is symmetric and positive definite.
SLIDE 39 Corollary 6. The tern (x(t), y(t), z(t)) ∼ N (0, Σ(t)) where Σ(t) solves dΣ(t) dt = A(t)Σ(t) + Σ(t)AT(t) + D(t)DT(t) Corollary 7. In particular for the global health indicator mσ
N we have
√ N(mσ
N(t) − mσ(t)) → N(0, Σx(t))
in particular P(mσ
N(t) ≥ M) ≈ N
√ Nmσ(t) − √ NM
→ The covariance matrix Σ(t) gives a measure of the volatility on the market. → These results (in particular Corollary 7) allow to characterize the distribution of aggregate losses in a large portfolio.
