Page 1

ssrn.com/abstract=1522544

Chaos and Bifurcation in 2007-08 Financial Crisis

Abstract

The impact of increasing leverage in the economy produces hyperreac-

tion of market participants to variations of their revenues. If the income

of banks decreases, they mass-reduce their lendings; if corporations sales

drop, and due to existing debt they cannot adjust their liquidities by fur-

ther borrowings, then they must immediately reduce their expenses, lay

off staff, and cancel investments. This hyperreaction produces a bifurca-

tion mechanism, and eventually a strong dynamical instability in capital

markets, commonly called systemic risk. In this article, we show that

this instability can be monitored by measuring the highest eigenvalue of

a matrix of elasticities. These elasticities measure the reaction of each

sector of the economy to a drop in its revenues from another sector. This

highest eigenvalue — also called the spectral radius — of the elasticity ma-

Keywords systemic risk, systemic crisis, econophysics, macroeconomics,

bifurcation, system stability, chaos

contagion of risks which is better described by the butterfly effect, which started

regularly being mentioned after the series of financial “unthinkables” that took

place in September of 2008, starting with the nationalization of government

sponsored enterprises Freddie Mac and Fannie Mae, the demise of the invest-

ment bank Lehman Brothers a week later, the fire-sale of another one Merrill

Lynch to Bank of America on the same day, and the government bail-out of the

insurance giant AIG just two days later.

Page 2

ssrn.com/abstract=1522544

“Butterfly effect” is a term used to describe a phenomenon such that small

changes at the initial stage result in a huge difference in long-term behavior.

The current financial crisis started in the U.S. real estate market and spread

to all over the world, and people are still debating when and how this crisis

will be over. Such a phenomenon is formally defined in dynamical systems as

sensitive dependence on initial condition. When a dynamical system possesses

a sensitive dependence on initial condition together with cyclical behavior, the

system often exhibits chaos, de Melo and Palis (1982).

In dynamical systems theory, a bifurcation refers to a structural modifica-

tion of the system behavior upon a continuous change in the parameters of its

equations. A catastrophe occurs when, following a bifurcation, a small change

in parameters discontinuously alters the equilibrium state of the economy. Dur-

ing the 2007-08 crisis, we did observe such a catastrophic event, where a mild

evolution of economic parameters ended into a drastic shift in financial interac-

tions. Before the 2007 subprime crisis, the economy was in what physicists call

a “meta-stable equilibrium”, that is, an equilibrium state that is destroyed by

a very small perturbation – like a dry forest totally burning upon the scratch

of a match – leading to a series of catastrophic events, until another basin of

attraction is reached, i.e. another stationary evolution mode, another cycle or,

even, a strange attractor as chaos theory predicts.

In this paper we suggest that the current financial crisis was mainly caused

by a breakdown of the dynamic stability of the financial system, according to

some catastrophic mechanism. More precisely, we start from a mathematical

model in Rn(the dimension n will be specified in the next section) of the finan-

cial system that exhibits a stationary state equilibrium. The financial activities

are considered as continuous perturbations of this equilibrium: when the per-

turbation is small enough, the equilibrium persists and the economy remains

stable. When the perturbation is too big, the equilibrium collapses and a finan-

cial crisis emerges. Furthermore, we show that the critical size of perturbations

that destroy the equilibrium shrinks when financial actors react more rapidly

and intensely to other actors they are in business with, leading to a meta-stable

equilibrium and a catastrophe. The critical perturbation size is directly related

to the debt and borrowing capacity, the leverage, and the market liquidity. In

other words our mathematical model shows the causal relation between leverage

Based on these observations, we propose the principles of methodology to

build an early indicator of the global system instability. The details of such

indicator still need to be worked out and tested, as all economic indices involved

in this methodology are not readily available.

In Section 2 we provide an intuitive view of the chaos in the current financial

crisis and relevant mathematics background. Section 3 will be fully devoted to

the structural stability and perturbation analysis of the financial system.

Page 3

ssrn.com/abstract=1522544

2Glimpse of Chaos

Financial crisis is generally defined as a situation in which some financial insti-

tutions or assets suddenly lose a large part of their value. The current (2007 -

2009+) crisis started in a small sector of global economy called “the U.S. real

estate market”. In the United States housing bubbles started to form due to low

domestic interest rates and the trade deficit which resulted in large foreign cap-

ital inflows. These two factors made easy and inexpensive credit available, and

many people started investing in real estate. The Case-Shiller Home Price Index

had its peak in the second quarter of 2006 [14], and the U.S. house prices have

steadily decreased since. However it is not only the U.S. houses that lost value.

Many banks and financial entities, both regional and global, went bankrupt.

Many companies, both small and large, went under as well. In both cases the

main cause was solvency and liquidity. During the development of the crisis,

the damage seemed to be only getting more severe, massive, and unpredictable.

Third year in the crisis, the current situation seems to be stabilizing, but the

future still looks unpredictable. In the meantime, the blame has been aimed at

financial engineers (also known as “quants”) for having created esoteric financial

derivatives and used faulty mathematical models to evaluate them.

Avoiding the question of model validity, which appears to us as a side ques-

tion, we try to understand the financial, then economic crisis in its dynamical

aspects.

Here we get a glimpse of chaos: what started locally has spread globally

with unpredictable severity, and this suggests sensitive dependence on initial

condition; the mathematical models which used to work well in the past do not

work all of a sudden, and this hints bifurcation of the system.

Typically chaos is found in dynamical systems that possess nontrivial recur-

rence (i.e. which cannot be isolated), and indeed there is “recurrence of risks”

behind the current financial crisis. In dynamical systems theory recurrence is

produced by the feedback loop, which in finance became global due to secu-

ritization. Although the original purpose of the securitization was to diversify

default risk, this “originate to distribute” practice spawned too many risky loans

which were destined to default. As a result the risk was disseminated globally as

opposed to diversified, then boomeranged back to the issuer of the loans as well

Page 4

during the real estate boom, cash flows that one usually calls “fixed” are the

scheduled ones, such as salaries, contributions to (pension and other invest-

ment) plans, coupons, installments, etc. Other cash flows are said “variable”

because they are at will: investments, loans, dividends, savings, etc. In fact,

both types of cash flows are impacted by economic conditions, the less variable

ones not necessarily being those called “fixed”. For example, the cash flow from

households to industry in exchange of goods and services remains almost con-

stant regardless of the economic condition. Noninterest income for banks such

as ATM fees and credit card fees are not affected by economic deterioration,

for banks can always raise those fees. Such cash flows are rather steady, while

payments from subprime loans, which are subject to default, are in practice

more variable cash flows.

At a macroeconomic level, variations of aggregate cash flows could be ex-

plained by shifts in the classical Hick’s IS-LM curves. However in order to

assess the actual market stability, we do not deduce variations of cash flows

from a model, but from empirical observations. Market instability, which is our

core target of study, may possibly result from a behavior that is predicted by a

model, e.g. Grandmont (1985), but it may also well be the consequence of the

economy departing from classical models.

Let us now broadly divide the economy into several segments in the spirit

of [5]. Typically home buyers (HB), which we do not distinguish from general

consumers, get financing from local mortgage lenders (ML) which include the

mortgage divisions of large banks. To facilitate financing with limited fund,

these mortgages are sold to large banks (LB — not only banks but other fi-

nancial institution that function like banks, for example brokers and insurance

companies) and the government sponsored enterprises (GSE — Fannie Mae and

Freddie Mac) for securitization — they are sliced, diced, and repackaged as mort-

gage backed securities (MBSs) which is a special kind of asset backed securities

(ABSs) or collateralized debt obligations (CDOs). Part of these MBSs are kept

within the banks (super-senior tranches) or are securitized again (ABS-squared)

and sold in secondary mortgage market. They are also sold to investors. The

investors (I) consist of many funds from all over the world, such as pension

funds, mutual funds, academic endowments, state employees’ retirement funds,

sovereign funds etc. Many people invest, directly and indirectly, in such funds,

to understand the dynamics of the free market. Government actions, when they

(i) HB to HB: home buyers invest in houses, and sell those to one an-

other.

(ii) C to C: companies invest into each other.

(iii) I to LB, GSE, C: investors buy stocks of LB, GSE, and C.