Systemic Risk and the Mathematics of Falling Dominoes

Reimer Kuhn

Disordered Systems GroupDepartment of Mathematics, King’s College London

http://www.mth.kcl.ac.uk/∼kuehn/riskmodeling.html

Teachers’ Conference, KCL, June 22, 2011

The Laws of Falling Dominoes

• A domino falls, if kicked sufficiently vigorously.

• A domino can be toppled by another domino.

• Avalanches can occur, if dominoes are set too closely.

2/23

Risk and Falling Dominoes

Operational Risk

Domino Theory & Spread

of Communism

Financial CrisisBlackouts in Power Grids

3/23

Fundamental Problem of Risk Analysis

• Estimation of risk

– Market: potential negative fluctuation of portfolio-value

(stock-prices, exchange rates, interest rates, economic in-

dices)

– Credit: potential change of credit quality, including de-

fault (asset values of firms, ratings, stock-prices)

– Operational: potential losses incurred by process failures

(human errors, hardware/software-failures, lack of com-

munication, fraud, external catastrophes)

– Liquidity: potential losses incurred by rare fluctuations

in cash-flows, requiring short term acquisition of funds to

maintain liquidity

4/23

Task

• Estimate PDF of

– Portfolio value at time T , PV (T), given PV (0)

– Market value of assets of obligors at time T , Ai(T), given

Ai(0)

– Losses L(T) due to process failures incurred during risk

horizon T

– Losses L(T) incurred during risk horizon T by need to

maintain liquidity in situations of stress

5/23

• Quantity of interest: Value at Risk

P(L)

L

1−q

EL

VaRq,T = (Qq[L(T)]−EL) e−rT Prob(L(T) ≤ Qq[L(T)]) = q

Money to set aside now, to cover losses in excess of expected

loss at time T , which are not exceeded with probability q.

• Requires good knowledge of tails of loss distributions.

6/23

Our Main Interest and Concern: Interactions

• Traditional approaches treat risk elements as independent or

at best statistically correlated

• Misses functional & dynamic nature of relations:

terminal–mainframe/input errors–results/manufacturer–supplier

relations . . .

• Effect of interactions between risk elements

- Possibility of avalanches of risk

events (process failures, defaults)

- Fat tails in loss distributions

- Volatility clustering in markets

(intermittency)

7/23

The Case of Operational Risks — Interacting Processes

• Conceptualise organisation as a network of processes

• Two state model: processes either up and running (ni = 0)

or down (ni = 1)

• Reliability of processes and degree of functional interdepen-

dence heterogeneous across the set of processes (quenched

disorder); connectivity functionally defined

⇒ model defined on random graph

• losses determined (randomly) each time a process goes down

(annealed disorder)

8/23

Dynamics – Mathematics of Falling Dominoes

• processes need support to keep running (energy, human re-

sources, material, information, input from other processes,

etc.)

• hit total support received by process i at time t

hit = ϑi −∑

j

Jijnjt + ηit

with ηit random (e.g. Gaussian white noise).

• process i will fail, if the total support for it falls below a

critical threshold (domino falls, if kicked too strongly)

nit+∆t = Θ

∑

j

Jijnjt − ϑi − ηit

9/23

Probability that a Domino Falls

• Probability of failure/probability of domino falling

Prob(

nit+∆t = 1∣

∣

∣n(t))

=∫

∑

jJij njt − ϑi

−∞dη p(η) ≡ Φ

∑

j

Jij njt − ϑi

p(η)

∑

jJij njt − ϑi

η

• unconditional and conditional probability of failure

pi = Φ(−ϑi)

pi|j = Φ(Jij − ϑi)

⇒ ϑi = −Φ−1(pi) , Jij = Φ−1(pi|j) − Φ−1(pi)

10/23

All-to-All Interactions — Mean Field Theory

• look at case with all-to-all couplings, (N ≫ 1)

Jij =J0

N, ∀ (i, j) ⇒

∑

j

Jijnjt =J0

N

∑

j

njt = J0mt

• Dynamics

nit+∆t = Θ

∑

j

Jijnjt − ϑi − ηit

= Θ(J0mt − ϑi − ηit) .

Thus (by LLN)

mt+∆t = =1

N

∑

i

Θ(J0mt − ϑi − ηit)

=1

N

∑

i

Φ(J0mt − ϑi) =⟨

Φ(J0mt − ϑ)⟩

ϑ

11/23

Mean Field Theory — Graphical Analysis

• Iterative dynamics mt+∆t =⟨

Φ(J0mt − ϑ)⟩

ϑ

mt

m t+1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 4 6 8 10 12 14 16 18

m

J0

Left: Graphic representation of iterated dynamics for three different values of J0 (curves

from left to right correspond to decreasing J0). Right: Location of fixed points as a

function of J0, for nonrandom ϑ = 2.25 ↔ pi ≃ 0.015. Backbending part of curve is

unstable fixed point.

12/23

Analysis for a Stochastic Setting

• Interactions on a random graph

Jij = cij

(

J0

c+

J√c

xij

)

with

P(cij) =c

Nδcij,1 +

(

1 − c

N

)

δcij,0 ,

symmetric (cij = cji), and

xij = 0 , x2ij = 1 , xijxji = α .

Study the limit

c ≫ 1 , N ≫ 1 .

Processes/organizations: small J ! 13/23

Results for α = 0

• Use limit theorems (LLN, CLT) to show∑

j Jijnjt is Gaussian.• If noise ηt ∼ N (0, σ) (also Gaussian), then

mt+∆t =

⟨

Φ

J0mt − ϑ√

σ2 + J2mt

⟩

ϑ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2 0 2 4 6 8 10 12 14 16

m

J0

2

4

6

8

10

12

14

16

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

J 0c

p

N at 0.035,8C at 0.025,5O at 0.015,3

O

C

N

(K Anand and RK, Phys Rev E75 (2007))

Left: Stationary fraction of down-processes for pi = 0.01, α = 0, and J = 0.2. Right: phase

diagram.14/23

Stationary Solution at α 6= 0

• Have to work much harder.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

m

J0

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16m

J0

(K Anand and RK, Phys Rev E75 (2007))

Stationary fraction of down-processes for α = 0.5 (left) α = 1.0 (right) at J = 0.2

15/23

Spontaneous Blackouts

• If mutual dependence of processes exceeds a certain level,

spontaneous blackouts can occur.

0 10000 20000 30000 40000 50000time

0

100

200

300

loss

es

0 10000 20000 30000 40000 50000time

0

100

200

300

loss

es

Loss record for a system of N = 50 interacting processes. Left panel: Coexisting low-loss

and high-loss phases @ pmaxi = 0.02 and (pi|j/pi)

max = 2.6. Right panel: Low-loss phase

simulated for the same pmaxi but (pi|j/pi)

max = 3. The low-loss situation is only meta-stable

and spontaneously deays to a high-loss situation via a bubble-nucleation process.

16/23

Key Features

• For sufficiently strong cooperative interactions — “dominoes

too densely placed” — get a (first order) phase transition:

coexistence of ‘functioning state’ and state of ‘catastrophic

breakdown’

• There is a critical Jc inside the coexistence region, beyond

which the system — while still working — prefers breakdown,

• Critical point at sufficiently high pi.

• Of special relevance: resilience to (external) stress can be

tested.

17/23

The Case of Credit Risks — Interacting Companies

• Risk arising from the possibility of obligors going bankrupt

or from changes in ‘credit quality’ (⇒ credit trading)

• Here only influence of defaults

• Two state model: company up and running (ni = 0) or down

(ni = 1)

• Probabilities of default and mutual impacts of defaults het-

erogeneous across the set of firms (quenched disorder); con-

nectivity functionally defined

⇒ model defined on random graph

• Losses determined (randomly: recovery process) when a com-

pany defaults (annealed disorder)

18/23

Dynamics — Falling Dominoes Again

• Companies need “orders” (support, cash inflow) to maintain

wealth and avoid default

• hit total wealth of company i at time t,

hit = ϑi −∑

j

Jijnjt − ηit

with ηit random (e.g. Gaussian white noise).

• company i defaults, if the total wealth falls below zero

nit+1 = nit + (1 − nit)Θ

∑

j

Jijnjt − ϑi + ηit

• No recovery within ‘risk horizon’ T : ni = 1 is absorbing state.

Time unit: 1 month; T = 12 ⇔ 1 year.

19/23

Global Analysis

• Analysis in a stochastic setting on random graphs as for OR,

• Loss distribution through fluctuating macro-economic

conditions. Noise model (minimal Basel II):

ηit =√

ρi η0t +√

1 − ρi ξit

with η0t slow economy-wide component, and ξit idiosyncratic

noise, and ρi = ρ(ϑi)..

〈nt+1(ϑi)〉 = 〈nt(ϑi)〉 +(

1 − 〈nt(ϑi)〉)

Φ

J0mt +√

ρi η0 − ϑi√

1 − ρi + J2mt

mt+1 = mt +

⟨

(

1 − 〈nt(ϑ)〉)

Φ

J0mt +√

ρ η0 − ϑ√

1 − ρ + J2mt

⟩

ϑ

20/23

Results CR — Defaulted Fraction and Loss Distribution

0

0.005

0.01

0.015

0.02

0.025

0 2 4 6 8 10 12

mt

t

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

-20 0 20 40 60 80 100 120 140 160 180

P(L

)

L

(JPL Hatchett and RK, J Phys A39 (2006))

Fraction of defaulted firms for neutral macro-economic conditions η0 = 0 at (J0, J) =

(0,0), (1,0), (0,1) and (1,1) (left, bottom to top; Loss distribution for a system with (J0, J) =

(0,0) and (1,1) and ℓ(ϑ) = 1/(ε + pd(ϑ)) (right)

21/23

Key Features

• Interpretation of parameters ϑi and Jij in terms of uncondi-

tional and conditional default probabilities

• For sufficiently strong interactions — in particular coopera-

tive interactions — get possibility of collective acceleration

of default rates in times of economic stress

• Typical behaviour less sensitive to interactions than rare events

⇒ fat tails in loss distribution; important for risk analysis

where rare event asymptotics is relevant.

• Due to initial conditions ni0 ≡ 0 and absorbing state:

⇒ no equilibrium dynamics

22/23

Summary

• Standard risk models based on statistically correlated risk el-

ements miss dynamically generated functional correlations

• Interacting processes capture functional dependencies in OR.

• Similarly: economic interactions ⇒ credit contagion in CR

• Physics analogy: lattice gas model on (random) graph.

• Describes bursts and avalanches of risk events (dominoes)

• Important: coexistence with phases of catastrophic break-down — no noticeable precursors !

• Generalized Credit risk model to include derivative trading (CDS).

• Neuro-dynamics model for intermittent dynamics & fat-tailed loss distributions in MR

23/23

THANK YOU

Supplementary Material

Results CR — Value at Risk

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

< L

>/<

L >

0 V

aR/V

aR0

R

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

< L

>/<

L >

0 V

aR/V

aR0

R

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

< L

>/<

L >

0 V

aR/V

aR0

R

(JPL Hatchett and RK, J Phys A39 (2006))

Ratios of value at risk (upper curves) and average losses (lower curves) for systems with

and without functional interaction evaluated along straight lines in the J0-J plane, with

R =√

J20 + J2. Left: J0 = 0 ; right: J = 0; lower: J0/J = 1.

Credit Risk with CDS

Fee payment

Reimbursement

Credit event

Protection buyer

Protection seller

Reference entity

paymentInterest

Credit

Fee payment

Reimbursement

Credit event

Protection buyer

Protection seller

Reference entity Bank

Credit

Interest payment

Left: Mechanics of a CDS used for hedging. Right: Mechnanics of a speculative CDS.

0.0001

0.001

0.01

0.1

1

10

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

P(L

)

L

0.0001

0.001

0.01

0.1

1

10

100

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

P(L

)

L

Baseline-scenario (red full line), and either one third (green long-dashed) or two thirds (blue

short-dashed) of the avarage exposure to firms are hedged by CDS, with insurance sector ,

Right: baseline-scenario (red full line) compared with situations where banks have completely

hedged their exposures through CDS and used this to either double (green long-dashed) or

triple (blue short-dashed) the size of their loan books.

Functional Dependencies in Market Risk

• Standard approach: Geometric Brownian motion (GBM) for

Log-returns of risk elements i (stocks, bonds . . . )

1

Si(t)

dSi(t)

dt= µi + σiηi(t)

• Statistical dependencies via correlated Gaussian noises

〈ηi(t)ηj(t′)〉 = ρijδ(t − t′)

• Is basis of JP Morgan’s Risk MetricsTM tool!

• But: no functional dependencies, no collective behaviour, no

market bubbles, crashes

28/23

Functional Dependencies in Market Risk — iGBM

• Functional dependencies between risk elements e.g. via

recommendations of analysts, economic dependencies.

• GBM in terms of hi(t) = log(Si(t)/Si0)

dhi(t)

dt= µi −

1

2σ2

i + σiηi(t)

• Minimal interacting generalisation (iGBM): stabilisation and

non-linear feedback (e.g. g(x) = tanh(x))

dhi(t)

dt= −κhi(t) + µi −

1

2σ2

i +∑

j

Jijg(hj(t)) + σiηi(t)

• ⇔ dynamics of of graded-response neurons

many meta-stable states; transitions between them ⇒ inter-

mittent dynamics.

Functional Dependencies in Market Risk – iGBM

• Solve in stochastic setting as for OR/CR

• Here MC study; trigger transitions by ‘unexpected news’

0 1000 2000 3000 4000 5000t/τ

−1500

−500

500

1500

∆I(t

)

−6 −4 −2 0 2 4 6∆h/σ

0.0001

0.001

0.01

0.1

1

10

σ P

(∆h/

σ)

τ=25τ=50

(RK and P Neu, J Phys A41 (2008))

Change of index I(t) = N−1∑

iSi(t) over time increment τ = 25 (left) and normalised

distribution of log-returns for τ = 25, and τ = 50 (right).