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.
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.
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 only 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.
... 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.
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¨ ollmer (1974), F¨ ollmer & 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?)
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 other agents. In particular: U ( ω i ) = u ( ω i ) + S ( ω i , ¯ m i ) + ǫ ( ω i ) . where ¯ m i is the subjective expected value of the choices of the other agents. → Under some conditions multiple equilibria are found.
Contagion in credit risk 1) Counterparty risk, Jarrow and Yu (2001) The default of a firm’s counterparty affects its own probability of default. 1 λ A hP ( τ A ≤ t + h | τ A > t ) = b 1 + b 2 1 { t ≥ τ B } . t := lim h → 0 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 λ A t = 1 { t ≤ τ A } h A ( Y t , Ψ t ). 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”).
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)
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.
Our goals 1. The credit quality of a firm is directly influenced by the state of other 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)
The model N firms active on the market. 1. The credit quality itself of firm i is described by a binary variable σ i ∈ {− 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 obligations.
The model N firms active on the market. 1. The credit quality itself of firm i is described by a binary variable σ i ∈ {− 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 obligations. 2. Strength of firm i described by a binary variable ω i ∈ {− 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
Time evolution and vehicle of contagion → To determine the time evolution on a generic interval [0 , T ] of the ( σ i ( t ) , ω i ( t )) i =1 , ··· ,N ∈ D 2 N [0 , T ] we “state” of the system, i.e. need to specify the rates/intensities for the transitions σ i → − σ i , ω i → − ω i . • Mean-field assumption: We let the interaction depend on the global health indicator (endogenous global factor) N N ( t ) := 1 m σ � σ i ( t ) N i =1
Time evolution and vehicle of contagion → To determine the time evolution on a generic interval [0 , T ] of the ( σ i ( t ) , ω i ( t )) i =1 , ··· ,N ∈ D 2 N [0 , T ] we “state” of the system, i.e. need to specify the rates/intensities for the transitions σ i → − σ i , ω i → − ω i . • Mean-field assumption: We let the interaction depend on the global health indicator (endogenous global factor) N N ( t ) := 1 m σ � σ i ( t ) N i =1 • The vehicle of contagion is given by m σ → → → ω j ω i σ i N fundam. indic. rating class global health indic.
Transition intensities for the system λ i := e − βσ i ω i , σ i → − σ i β > 0 with intensity µ j := e − γω j m σ ω j → − ω j N , γ > 0 with intensity → 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 m σ N is higher. → May be generalized to letting the rates depend on exogenous macroeconomic factors. (business cycles)
→ The state variables form a continuous-time Markov chain on the configuration space {− 1 , +1 } 2 N with infinitesimal generator L acting on f : {− 1 , 1 } 2 N → R as N � f ( σ i , ω ) − f ( σ, ω ) � � Lf ( σ, ω ) = + λ i i =1 N � � f ( σ, ω j ) − f ( σ, ω ) � + µ j j =1 where σ i = ( σ 1 , ..., σ i − 1 , − σ i , σ i +1 , ..., σ N ) ; analogously for ω j .
General results Concern the dynamics of the system ( σ i ( t ) , ω i ( t )); i = 1 , ..., N and of m σ 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.
A crucial object: The empirical measure ρ N ∈ M 1 ( D 2 [0 , T ]) . N ρ N = 1 � δ { σ i [0 ,T ] ,ω i [0 ,T ] } N i =1 for a generic function f of the trajectory one has N 1 � � f ( σ i [0 , T ] , ω i [0 , T ]) = fdρ N N i =1 Remark: Derive the empirical means: � fdρ N = m σ if f ≡ Π t ( σ i [0 , T ]) then N ( t ) � fdρ N = m ω if f ≡ Π t ( ω i [0 , T ]) then N ( t ) � fdρ N = m σω if f ≡ Π t ( σ i [0 , T ] · ω i [0 , T ]) then N ( t )
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