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The Reasonable Effectiveness of Mathematics in Economics We are now in the midst of the worst economic crisis since the Great Depression. What had started as the subprime crisis was followed by large losses at investment banks and hedge funds, falling equity markets, massive job losses and corporate bankruptcies. Given an (apparently) collective failure to foresee (i.e., forecast) events, where does this leave the dismal science of economics and financial modeling?
In The Reasonable Effectiveness of Mathematics in Economics, Sergio Focardi and Frank Fabozzi argue that economics is indeed a science, and that economic and financial models have worked reasonably well. According to the authors, the fault lies not so much with our science as with the economy and financial markets as we have designed them. The present situation calls for the rationality that comes from the discipline of empirical science and the rigor of mathematical language. Models in themselves have not created this crisis and, if properly used, they might help us find reasonable solutions. However, in using our models as scientific tools we should understand that the models we use cannot be based on the same mathematics as models built to describe stable physical realities.
Our economic and financial models extract only a small quantity of information from a large amount of noise. This situation has no parallel in physics, where we have a large amount of information corrupted by a small amount of noise. Perhaps the modeling techniques used in biomathematics or in mathematical ecology are more appropriate. Here is why. Financial models must be adaptive and cope with the problem of a fundamental scarcity of data. To the observer, scarcity of data produces randomness. We now have better tools to separate noise from information. We can compare empirical data with randomly generated data in a highly structured way. Methodologies such as Random Matrix Theory and Information Theory offer effective benchmarks to separate meaningful structures from random ones; they allow to establish random benchmarks for complex mathematical objects such as covariance matrices.
We also have better tools to reduce the dimensionality of our observations. We are now able to understand large factor models in a dynamic context while previously we were only able to understand either small dynamic models or large static models. We also have a better understanding of self-organization. A simple but effective example of self-organization is clustering. With clustering we form groups of similar objects, for example similar stocks. The organization of groups is not predetermined, as in standard industry classifications, but is self-created by empirical features such as return correlation or price cointegration.
In summary, our models work reasonably well, given the constraints of the design of financial and economic systems. But we have to use simple models suitable for highly noisy, complex systems.
Developed in “The Reasonable Effectiveness of Mathematics in Economics” Focardi and Fabozzi, forthcoming in The American Economist
Avoiding Black Swans The
year 2007 was not a good year for quantitative funds. The consensus is that
in July-August 2007 the subprime crisis created an unexpected liquidity
crunch that forced a number of highly leveraged quantitative hedge funds to
liquidate positions. This sudden and massive sale of assets provoked an
inversion of the behavior of factors used by many quantitative funds, leading
to sizable losses only partially recovered since. Quantitative funds and the
mathematical modeling of financial markets came under attack. Could
mathematics be used to describe financial markets given that markets are
subject to sudden and unpredictable changes that models estimated on past
data cannot predict?
Nassem
Talib coined the term “black swans” to denote sudden
unpredictable changes or events with major effects on markets and economies.
The existence of such events is undeniable. However, "black swans"
are not confined to economics; they also populate the world of the physical
sciences. Consider large-scale unpredictable events such as tsunamis,
earthquakes, or major epidemics. It is not at all certain that such events
can be predicted. Modern science has abandoned the idea of universal
determinism. Quantum mechanics, non linear dynamics, and the theory of
complex systems have forced scientists to accept the view that our science is
characterized by fundamental uncertainty at both the microscopic and the
macroscopic levels.
However,
we do not blame science for this uncertainty, nor do we reject mathematical
physics because major events are unpredictable. The principle of
indeterminacy of quantum mechanics and the theory of chaos have changed our
view of the world but we have not rejected mathematics in our exploration of
the physical world. On the contrary, we try to refine our knowledge and
better understand its limits. We adopt principles of safe design to
mitigate the consequences of uncertainty. For example, we reinforce houses in
seismic areas and do not fly planes in unpredictable turbulence.
Is
it so different with economics and finance? Black swans do exist but it is
likely that – wanting to - we could avoid most of them. Was the
subprime crisis unpredictable? Are large losses unpredictable when leverage
is in the range of 6-8 times? Perhaps, but it is surprising that the fat
tails of return distributions were so grossly underestimated.
As in mathematical physics, so in mathematical economics: we must be ready to change our paradigm. This will call for adaptive models, able to follow changing market conditions. Here is the problem. Adaptive models are by nature non linear and require lots of data to estimate, but data in finance is scarce so we need modeling ingenuity to create models that adapt using our limited data samples. We will probably also need to pay more attention to model risk.
Developed in “Black Swans and White Eagles: On Mathematics and Finance,” Focardi and Fabozzi, forthcoming in Mathematical Methods in Operations Research.
The
Magic of Momentum Strategies First reported in Jagadeesh and Titman (1993), momentum remains a
price anomaly difficult to explain. A distinction needs to be made between 1)
global models valid over long periods of time and 2) local models
approximately valid on only selected time windows and updated in real time.
The latter belong to the domain of real-time econometrics.
Momentum
and reversals, as described in Jagadeesh and Titman (1993) and subsequent
literature, are approximate local models of returns that call for real-time
reparametrization. In fact, the finding of momentum and reversals is
equivalent to the finding that:
If we change the length of the moving window, returns change. We can thus create a term structure of momentum/reversal returns in function of length of the moving time window. In the classical description of momentum/reversal, we have a different model for each time window.
Though
a global model able to represent the entire term structure of
momentum/reversal returns, compatible with other known stylized facts and
independent of the length of the estimation window is both theoretically and
practically important, from the point of view of portfolio formation with
momentum/reversal strategies, improving local models would represent a
significant advance in term of performance. To this end, the modeler can
chose from a large family of approximate local models that includes
autoregressive models and factor models that consider both risk adjustment
and dynamic correlations. In particular, introducing correlations and risk
measures allows for risk-return optimization, a significant advance over
classical momentum strategies. Optimization reduces model risk and allows to
reduce the turnover. Autoregressive models offer many advantages, including
the ability to exploit cross autocorrelations and cointegration phenomena,
adding an important source of profit.
Can We Make Fat Tails Work For Us? Gains realized in the equity markets since the beginning of 2006 have been virtually wiped out since the beginning of May. Such non normal but not-so-infrequent market movements are referred to as the "fat tails" of market return. The problem for asset managers is to avoid losses and, eventually, turn losses into gains. This calls for an understanding of the mechanisms that generate fat tails. While today’s linear models describe how fat tailed noise propagates in time and affects different securities, they cannot explain how fat tails are generated starting from Gaussian noise. Methods used to explain the generation of fat tails include neural networks, non linear regressions, and a modeling strategy that consists in coupling different models. Among the latter are the ARCH models, stochastic volatility models, and, more recently, regime shifting and Markov switching models. These models are intuitive: they implement the idea that fat tails are generated by a switch in economic states. Subordinated models are another elegant way to consider non linearities. Widely used in insurance problems, subordinated models capture the notion that fat tails are generated by changes in the market activity rate. However, using non linear models requires the estimation of a much larger number of parameters, and these parameters might be difficult to control: they are prone to generate chaotic movements.
Developed in Financial Econometrics : From Basics to Advanced Modeling Techniques Rachev, Mittnik, Fabozzi, Focardi, Jasic (Wiley, 2007)
Can We Trust Our Models? Can
large sums of money be confidently entrusted to computerized processes that
automatically select a mix of investments? The question is important for
those o manage assets as well as for the investor. The short answer is:
Yes, but with caveats. In “Implementable Quantitative Research” (The
Journal of Alternative Investments), Fabozzi, Ma, and The
Intertek Group partner Focardi show how quantitative research is performed
and converted into implementable trading strategies. The authors discuss
pitfalls to avoid in the model selection and validation processes and propose
methodologies for model estimation and risk control. In addition to a solid
theoretical basis behind models, the critical issue is how to mitigate model
risk.
Developed in "Implementable Quantitative Research" Fabozzi, Ma and Focardi, The Journal of Alternative Investments, Fall 2005
Explaining
Credit Risk Contagion Empirical studies show that credit events are subject to "epidemics" in the sense that prolonged periods when credit problems are rare are followed by prolonged periods when credit problems are frequent. This behavior is similar to the ARCH behavior of volatility. The behavior of credit risk can be ascribed in part to common factors, but chains of mutual interaction between firms also play a part. In “An Autoregressive Conditional Duration Model of Credit-Risk Contagion” (The Journal of Risk Finance, Fall 2005), Fabozzi and The Intertek Group partner Focardi discuss the applicability of the Autoregressive Conditional Duration (ACD) model to credit events. Introduced by Jeffrey Russell and Noble-prize winner Robert Engle, ACD models represent processes formed by single events that happen with variable frequency. The authors suggest that the ACD model provides a convenient framework for representing the time evolution of credit events at the aggregate level. The advantage of the ACD model is twofold: 1) it allows to represent the effect of both exogenous factors and mutual interactions on credit events and 2) it allows financial institutions to gain a quick but precise understanding of their credit risk exposures.
Developed in “An Autoregressive Conditional Duration Model of Credit-Risk Contagion” Focardi and Fabozzi, The Journal of Risk Finance, Fall 2005
It is now widely accepted that asset returns are not normally distributed. One aspect of the non-normal distribution of returns that has important implications for both portfolio management and risk management is the weight of the distribution tails, which dictates the likelihood of extremal events. Events that are negligible under Gaussian distributions (e.g., the six-sigma events) become critical if the tails have more weight. In "Fat Tails, Scaling, and Stable Laws: A Critical Look at Modeling Extremal Events in Financial Phenomena" (The Journal of Risk Finance, Fall 2003), Fabozzi and The Intertek Group partner Focardi discuss the modeling of extremal events. The authors show how different tail weights result in different behavior of the sum of variables (e.g, portfolio returns) and the maxima (informally defined as the largest value in a sample). Non-normality is not equivalent to fat tails but implies an accurate estimation of tail behavior. The authors discuss why different non-normal tail behavior requires fundamentally different modeling methods for risk and portfolio management.
Developed in "Fat Tails, Scaling, and Stable Laws: A Critical Look at Modeling Extremal Events in Financial Phenomena" Focardi and Fabozzi, The Journal of Risk Finance, Fall 2003
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