Capturing Low-Probability, High-Impact Events 'Black Swans' in Economic and Financial Models

Thesis / Decision-making and Valuation

Capturing Low-Probability, High-Impact Events 'Black Swans' in Economic and Financial Models
Jamilu Auwalu Adamu , Lecturer, Nigeria

Incorporation of Fat - Tailed Effects of the Underlying Assets Probability Distribution using Advanced Stressed Methods.

Capturing the effects of Low-Probability, High-Impact "Black Swans" in the existing stochastic and deterministic models is tremendously Important. On this page, I would like to share with the members an open access, peer reviewed published research findings of my PhD thesis on how to capture the effects of Low-Probability, High-Impact Events in our existing economic and financial models.
I shall begin with the incorporation of fat-tailed effects of the underlying assets probability distribution in the popular LOGIT and PROBIT MODELS.


The Global financial markets have experienced series of financial and economic crises right from the inception and from generation to generation. Banks, Companies and the world economy experienced catastrophic deterioration and serious corporate failures by systemic risk effect.
Big Banks and Companies like Continental Illinois, City Federal Savings and Loan, Bank of New England Boston, Lehman Brothers, General Motors, and Worldcom have all failed and declared bankrupt in the history of global financial markets. That is why, many scholars in the past and recent past have attempted to comes up with the models that can precisely calculate the Probability of Default of a Bank or Company over a given time period. Probability of Default for a given Company or Bank captures the probability that the Company or Bank will default within a certain period.
The most popular models used by financial institutions to calculate probabilities of default are LOGIT (1980) and PROBIT (1981). Despite the fact that Logit and Probit gives good approximations but seems not to capture chaotic markets behaviour to some extend. Accurately determination of probability of default play very important role in the entire world economy. Probability of Default is the major component when determining (i) Capital Requirements under Basel II (Now Basel III) (ii) Expected Loss (iii) Risk Weighted Asset.
Also, the probability of default (PD) is a crucial parameter in risk management, and can be used for the requests of loans, rating estimation, pricing of credit derivatives and many others key financial fields. The false estimation of PD leads to unreasonable rating; incorrect pricing of financial instruments and thereby it was one of the causes of the recent global financial crisis.
The aim of this research work is to come up with new advanced stressed probability models that can capture chaotic markets behaviour or in the other way round to work under financial markets distress to some extend.


Initially, corporate distresses were assessed based on some qualitative information, which were very subjective. In particular, four references were mostly used, namely: (i) the capacity of the manager in charge of the project or company, (ii) the fact that the manager had an important financial involvement in the company as a financial guarantee, (iii) the project and the industry in itself, and (iv) the fact that firm possessed assets or collateral to back-up in case of a bad situation.

John M. Moody (1909) was the first to published credit rating grades for publicly traded bonds. In 1941, David Durand applied discriminant analysis proposed by Fisher (1936) to classify prospective borrowers. Attempts have been made in 1950s to merge automated credit decision making with statistical techniques so as to enhance credit decision making. Lack of sophisticated computing tools, the models possessed limitations. Myers and Forgy (1963), compared discrimination analysis with regression in credit scoring application.

Altman (1960), introduced variables in a multivariate discriminant analysis and obtained a function depending on some financial ratios. Beaver in 1966 introduced an univariate approach of discriminant analysis to assess the individual relationships between predictive variables, and subsequent failure events. In 1968, Altman expanded the work of Beaver (1966) to allow one to assess the relationship between failure and a set of financial characteristics. Martin (1977), presented a logistic regression model to predict probabilities of failure of banks using data obtained from Federal Reserve System. Ohlson (1980), used Logit to predict bankruptcy. Zmijewski (1984) used probit to estimate probability of default and predict bankruptcy.

In 1985, West used factor analysis and logit estimation to assign a probability of a bank being a problematic. In 2001, Shumway introduced dynamic logit or hazard model to predict bankruptcy. Chava & Jarrow (2004), Hillegeist, Keating, Cram, & Lundstedt (2004), and Beaver, McNichols & Rhie (2005) uses Shumway’s approach. In 2004, Jones & Hensher introduced a mixed logit model for financial distress prediction and argued that it offers significant improvements compare to binary logit and multinomial logit models. Campbell, Hilscher, & Szilagyi (2008), introduced a dynamic logit model to predict corporate bankruptcies and failures at short and long horizons using accounting and market variables.

In 2011, Altman, Fargler, & Kalotay used accounting – based measures, firm characteristics and industry level expectations of distress conditions to estimate the likehood of default inferred from equity prices. Li, Lee, Zhou, & Sun (2011) introduced a combined random subspace approach (RSB) with binary logit models to generate a so called RSB-L model that takes into account different decision agent’s opinions as a matter to enhance results.
Sun & Li (2011) tested the feasibility and effectiveness of dynamic modelling for financial distress prediction (FDP) based on the Fisher discriminant analysis model.

Stefan Van der Ploeg (2011) stated that since the seminal work of Martin (1977), the Logit and Probit Models has become one of the most commonly applied parametric failure prediction models in the academic literature as well as the banking regulation and supervision. Jamilu (2015) introduced new methods entitled “Jameel’s Advanced Stressed Methods uses Jameel’s Criterion” to Stress Economic and Financial Stochastic Models, initially using Logit and Probit Models.


Before incorporating Fat - Tailed Effects in our Existing LOGIT and PROBIT Models. We have to discuss about how to obtain BEST FITTED FAT -TAILED PROBABILITY DISTRIBUTIONS OF THE UNDERLYING STOCKS RETURNS OF THE COMPANIES UNDER CONSIDERATION.

However, the major aim of this Post of the Research Findings is to consider Eleven (11) out of fifty (50) WORLD’S BIGGEST PUBLIC Companies by FORBES as of 2015 Ranking regardless of the platform in which they are listed and run the test of Goodness of fit on them vis – a – vis time series from 2014 – 2009. These include:

(1) China Construction Bank Corporation (CICHY) from 2014 – 2009
(2) Bank of China Limited (3988.K) from 2014 – 2009
(3) Berkshire Hathaway Inc. (BRK – A) from 2014 – 2009
(4) Toyota Motor Corporation (TM) from 2014 – 2009
(5) Volkswagen Group AG (VLKAY) from 2014 – 2009
(6) Bank of America Corporation (BAC) from 2014 – 2009
(7) Nestle India Limited (NESTLEIND.NS) from 2014 – 2009
(8) International Business Machines Corporation (IBM) from 2014 – 2009
(9) Goldman Sachs Group Securities (GJS) from 2014 – 2009
(10) Google Inc. (GOOG) from 7/2/2015 to 5/18/2012
(11) Facebook Inc. (FB) from 7/2/2015 to 5/18/2012

This is regardless of the questions of WHAT would happen to the Stock Returns probability distributions IF we considers:
(a) Daily, Weekly, Quarterly or Annually Stock Returns
(b) Companies that are more than Five (5)
(c) Companies that are listed on different stock exchanges not only ONE STOCK EXCHANGE platform
(d) Simultaneously Long – term and Short – term time series of the Stock Returns
(e) Recently past public companies like FACEBOOK with initial public offering on the 18th May, 2012 and began selling stock to the public and trading on the NASDAQ the same date and the GOOGLE on the March 9 2006.

To achieve these, the author developed what is called JAMEEL'S CRITERION using ESSAYFITS SOFTWARE as follows:


(i) We accept if the Average of the ranks of Kolmogorov Smirnor, Anderson Darling and Chi-squared is less than or equal to Three (3)
(ii) We must choose the Probability Distribution follows by the data ITSELF regardless of its Rankings
(iii) If there is tie, we include both Probability Distributions in the selection
(iv) At least Two (2) probability distributions must be included in the selection
(v) We select the most occur probability distribution as the qualify candidate in each case of test of goodness of fit of the stock returns as follows

In the next post, we shall continue from Jameel's Criterion.

Thank you.

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