Table of Contents

## The Chaos Theory

*“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.” *

## Albert Einstein

*This statement best describes the Chaos theory. The theory appreciates the complexity of nature, its unpredictability and its non-linear nature. While traditional science concentrated its efforts in definite measures of nature such as amount of rainfall, gravity, electricity, chemical reactions, they did not determine the complexity of the interactions of these systems *(Kellert 1993)*. For example, for one to predict the amount of precipitation, they must determine the effects of several weather factors such as winds, humidity, among others. This makes it difficult to predict the weather with precision. In business, experts try to predict customer behavior but they can only make estimations and not concrete determinations. Aspects such as landscapes, brain works, trees, rivers and general natural systems manifest very complex behavior. Fractal mathematics is a branch in mathematics that tries to capture the infinite, complex and chaotic behavior of nature *(Kyrtsou and Terraza 2003)*. *

Devaney (2003) defined a chaotic system as one whose future is affected by the current conditions, one that has topological mixing and one that has periodic orbits that are dense. Scientists used a simple illustration of a double rod pendulum to illustrate the complexity of nature, using the tip of the second rod to sketch its path (Kellert 1993). The resultant diagram is not only complex, but also shows how difficult it is to predict the next path that the tip would go through. While the path of the first rod is definite, an additional rod complicates the cycle thus creating ‘chaos’ in situations (Hristu-Varsakelis and Kyrtsou 2008).

Application of Chaos Theory in Management of Defaulting in Lending Institutions

The model has been used in predicting the probability of default for loan borrowers. The complex nature of humanity and possible inability to predict the future is equally similar in lending institutions. The institutions use chaos model to estimate the probability of default, PD. The role of attractors in the chaotic models is to create stability that provides a small ground under which the chaotic system can be estimated albeit with very little accuracy. Volatility is reduced as well as explosions that ensure that a certain degree of confined results is obtained. One of the fundamental characteristics of the chaotic theory is that the entire system is attached to the starting point. This can be elaborated using the map defined by:

Xt = 4Xt−1(1 − Xt−1), t = 1, 2, …

The solution to this indicates that the outcome of the system would be within the starting points of the system (Dominique 2009). If the starting point lies between 0 and 1, then the entire system would lie within these two points.

The above explanation is very important in application of PD. The starting point determines the level of risk that the lender subjects themselves to when they lend money to a borrower. Institutions ideally assess the current status of the borrower and determine whether to issue the money or not. Lyapunov exponents are among the most common measures that are used in this case (Dominique 2009). They use mathematical ideas to determine the orbits that confine the system through the attractors.

Lending is subject to many shocks and bumps due to the nature of the chaotic environment it operates. While some fluctuations are triggered by external forces that tilt the complex lending and repaying system, some are internal and hard to detect. Chaos can arise from different sources including traditional models such as cobweb models and asset pricing through the introduction of societal changes such as beliefs and practices.

It is hard to effectively determine the presence of attractors. With the knowledge of the current conditions of the borrower by highlighting their present financial ability, a positive Lyapunov exponent can be calculated. A phase space can also be computed through fractal dimension that is represented as dH,d>2dH+1. Once these two have been determined, successive embedding would be used to determine their positivity as well or apply the rather known time series method in a situation where the analytical system remains unknown (Dominique 2009). This way, the risk of giving out the loan can be determined. Some of the attractors used in PD are aspects such as the general flow of money by the borrower, their past lending history, use of collateral for the loans, guarantors among other strategies. All these ensure that the system is controlled and the starting point is known. The lender might also look into the general economic situation in the country or region and try to predict the future of the economic activity that the borrower would be involved in at the time of borrowing.

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Financial institutions appreciate that every lending is a risk to the lender because the future cannot be accurately predicted (Kyrtsou and Vorlow 2005). Such risk means that the lender faces a potential loss of both their principal and their interest. One of the models that they could employ to predict the future of lending is the chaos theory. Lending often faces many challenges because the money obtained is not always used for the purpose it was intended and the expected returns are highly diminished. Businesses are often crashed by changing macroeconomics in their respective countries which may not be foreseen by either the lender or the borrower. An example is the Greece situation where the economy collapsed and all loans and money held in trusts and other media was frozen. Other personal factors may also arise and disrupt the cash flow of the borrower, disabling them to continue repaying the borrowed money (Kyrtsou and Labys 2006). Business collapses and bankruptcies, health issues, natural disasters, among other factors have been cited as reasons for loan default in many countries. At times, the lender even miscalculates the ability of the lender who consequently borrows more money than they can service.