Table of Contents

## 1. Introduction

Bankruptcy prediction is a crucial issue in financial planning. If a bank or a company is about to go bankrupt, it becomes irrational to invest money in them. Moreover, it might be reasonable to withdraw current investments in order to avoid further losses. Although there is no absolutely reliable bankruptcy prediction method, the use of research-based forecasting models can greatly reduce the risk and uncertainty.

The aim of any bankruptcy prediction method is to be able to estimate a company’s financial prospects due to available data on its current performance with a sufficient degree of accuracy. Ribeiro et al. (2005) define that the bankruptcy prediction problem is “to discriminate between healthy and distressed companies based on the record of several financial indicators from previous years” (p. 419). The approaches to solution of this task fall into two main categories – conventional and modern methods. The first group includes discriminant analysis, multi-criteria analysis, and statistical methods, while the second one comprises neural networks and genetic algorithms (Thomaidis et al., 1999).

The research literature provides no generally accepted definition of conventional bankruptcy prediction methods. Most commonly, this term is associated with the multivariate discriminant analysis (MDA) model offered by Altman (1968) and logistic regression model proposed by Ohlson (1980). Both models used a company’s financial ratios as explanatory variables to determine the probability of failure (Jardin, 2008). Later, statistical models based on time series, such as ARIMA (Auto Regressive Integrated Moving Average), came into use.

This paper examines the three most common conventional methods for bankruptcy prediction – the MDA model, the logistic regression model, and the ARIMA model. It culminates in a conclusion on their comparative accuracy and efficiency.

## 2. Altman’s Multivariate Discriminant Analysis (MDA) Model

Altman’s method uses five financial ratios as independent variables in a linear model to calculate the probability of bankruptcy. The significant ratios are:

- Net working capital to total assets.
- Retained earnings to total assets.
- Earnings before interest and taxes to total assets.
- Market value of equity to the book value of total liabilities.
- Annual sales to total assets.

All figures for calculation of these ratios can be obtained from a company’s financial statements, which makes MDA model extremely practical and easy to use.

To determine significant ratios, Altman (1968) analysed a sample of 66 American industrial companies, half of which failed in the 1950s and 1960s while the other half did not. For each failing company, he had chosen a surviving one that matched it most closely by categorical characteristics, such as equity value and industry profile. The researcher analysed the differences in a large number of financial ratios between these pairs and found discrepancies in the five ratios listed above to be the greatest. They were included in the MDA model as the most reliable predictors of bankruptcy.

The model assigns each ratio a coefficient to account for its relative importance in bankruptcy prediction. The coefficients were first determined by Altman (1968) from his company’s sample. The resulting linear model is:

*Z *= *α _{1}x*

_{1}+

*α*+

_{2}x_{2}*α*+

_{3}x_{3}*α*+

_{4}x_{4}*α*, where

_{5}x_{5}Z is a company’s Z-score, an aggregated predictor of bankruptcy,

x_{1} … x_{5} are financial ratios, and

α_{1} … α_{5} are MDA coefficients.

Altman (1968) considered Z-score lower than 1.81 as an indicator of a high default risk. Z-score higher than 2.99 indicated a low bankruptcy risk, while scores between 1.81 and 2.99 fell into the “grey” or “ignorance” zone. In other words, the author of the model recognized that it could not accurately predict the risk of bankruptcy for this group of companies.

Although the MDA model has significant drawbacks – in particular, it does not account for ratio dynamics and their possible interdependence – it is still frequently used by researchers. A probable explanation of this ongoing popularity is the model’s extreme simplicity and flexibility combined with a rather high degree of accuracy. For example, in Karamzadeh’s (2013) study of 90 Iran companies, half of which later went bankrupt, Altman’s model predicted bankruptcy in 74.4 % of cases in one-year advance and in 64.4 % of cases in two-year advance.

The original MDA coefficients are seldom used by present-day researchers. Most of them determine these coefficients specifically for their sample to improve the model’s predicting power in the context of their industry and market situation. In particular, Grice and Ingram (2001) pointed out the need to re-estimate bankruptcy model coefficients even for the same industry over time. Other researchers also alter the number of variables to meticulously describe a particular industry. For example, Kedar-Levy et al. (2014) in their study of highly-leveraged Israeli firms used an adapted Altman’s model with a new ratio (EBITDA to paid financing expenses) added. The coefficients applied were determined by an earlier researcher specifically for Israeli markets, which use different accounting rules than in the US (Kedar-Levy et al., 2014, p. 3). The flexibility and adaptability of Altman’s model makes it suitable for bankruptcy prediction in all kinds of companies and industries.

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## 3. Logistic Regression Model

Ohlson (1980) offered his logistic regression model as an alternative to Altman’s, being concerned about the statistical restrictions imposed by that model. Ohlson’s model shares the underlying principle of linear regression, but uses nine independent variables, some of which are binominal:

- SIZE = Log (Total assets/GNP price-level index)
- TLTA = Total liabilities/Total assets
- WCTA = Working capital/Total assets
- CLCA = Current liabilities/Current assets
- OENEG: 1 if total liabilities exceed total assets, zero otherwise.
- NITA = Net income/Total assets
- FUTL = Operational funds/Total liabilities
- INTWO: 1 if net income was negative for the last two years, zero otherwise.
- CHIN = (NI
_{t}– NI_{t-1})/( │NI_{t}│+│NI_{t-1}│), where NI_{t}is the net income for the last year.

All these variables are readily obtained or derived from published corporate financial statements, which make Ohlson’s model easy to use and suitable for any kind of research.

The model uses coefficients that were estimated by Ohlson (1980) from a pool of 2000 companies and employed by most subsequent researchers. The resulting linear equation is:

*O*=–1.32–0.407*X _{1}*,–6.03

*X*–1.43

_{2}*X*–0.0757

_{3}*X*–2.37

_{4}*X*–1.83

_{5}*X*–0.285

_{6}*X*–1.72

_{7}*X*–0.521

_{8}*X*

_{9 }, where

O is the dependent variable (O-score),

X_{1} … X_{9} are the independent variables listed above.

To evaluate the probability of bankruptcy, exp(O-score) is divided by (1+exp(O-score)). If the resulting value is 0, the company is likely to fail in two years; if the resulting value is 1, failure is unlikely. As the probabilistic measure is binominal, the zone of uncertainty is eliminated.

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Today, researchers often use the logistic regression model because of its practicability and rather high predicting power. Some even claim Ohlson’s model to be the most accurate among those using the logit regression principle. However, the research evidence on its practical implementation is mixed. In a study of 60 companies listed on Tehran Stock Exchange by Jouzbarkand et al. (2013), Ohlson’s model accurately predicted 91.7 % of failures (p. 155). Meanwhile, in Karamzadeh’s (2013) study of 90 corporations from the same stock listing, this method only predicted 53.3 % of bankruptcies in one-year advance and 64.4 % – in two-year advance (p. 2010).

Overall, Ohlson’s logistic regression model is much similar to Altman’s in both advantages and disadvantages. It also uses a set of financial ratios as independent variables in a linear model to estimate the probability of corporate failure. Ohlson’s model accounts for the time dynamics – in particular, it measures the difference in net income. The main drawback of this method is that it rigorously divides all companies into two groups – likely and unlikely to fail – instead of regarding them as more and less likely to fail. However, since bankruptcy is a time-evolving event (Doumpos & Zopounidis, 2006, p. 185), its likelihood is changing over time. Hence, it is more useful to consider it in a dynamic context.

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## 4. Time Series Forecasting Models

Time series forecasting models, sometimes referred to as the Box-Jenkins methodology, rely on a historical pattern of data to predict its future values. One of their most common applications is aimed to determine the bankruptcy problem as they are known to be very powerful for short-term forecasts (Shim, 2000, p. 103). Time series models use either past values (autoregressive models, AR) or past errors (moving average models, MA). The most advanced models, known as ARIMA, combine both these factors.

For various kinds of data, different kinds of time series models can yield the most accurate results. To select the most relevant model for each case, a researcher needs to apply identification fitting and diagnosis checking (Lawrence & Klimberg, 2014, p. 31). At the first stage, a sample of simple autocorrelation functions is generated from the data and compared to a theoretical set of such functions of the known forecasting models. The closer the actual distribution is to the theoretical one, the better this model fits the data. Other things being equal, a model with a smaller number of coefficients is considered more appropriate. At the second stage, the coefficients are estimated. They should yield a minimal sum of squared residual errors. At the third stage, the research examines residual errors. Their samples should be normally distributed with mean zero. These three stages should be repeated for every kind of model – AR, MA, and ARIMA – to select the most appropriate one for this data set.

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To use ARIMA for bankruptcy prediction, the time series data on the company’s financial performance should meet the stationarity criterion. Stationary data fluctuates around a constant mean or variance with no trend over time, being not subject to seasonal changes (Shim, 2000, p. 104). As real-world financial data typically shows upward or downward trends, for the purpose of ARIMA analysis it needs to be differenced as ΔY_{t} = Y_{t} – Y_{t-1} , where Y_{t} is the financial indicator value in the last period, and Y_{t-1} – in the preceding period.

The general view of ARIMA model is:

*Y _{t}*=

*a*+

_{0 }*a*+

_{1}Y_{t-1 }*a*+...+

_{2}Y_{t-2 }*a*+

_{p}Y_{t-p}*ε*+[–

_{t }*b*–

_{1}ε_{t-1}*b*– ... –

_{2}ε_{t-2}*b*] , where

_{p}ε_{t-p}Y_{t} is the response variable at time t,

Y_{t-1}, Y_{t-2} … Y_{t-p} are response variables at various time lags,

a_{0}, a_{1} … a_{p} are the coefficients to be estimated,

ε_{0} is the error term (variables not explained by the model),

ε_{t-1}, ε_{t-2} … ε_{t-p} are the errors at various time lags,

b_{0}, b_{1} … b_{p} are the coefficients to be estimated (Lawrence & Klimberg, 2014, pp. 32-33).

As ARIMA model combines the features of other two kinds of time series models, accounting for both past values and past errors, it usually yields the most accurate estimates of bankruptcy probability.

In bankruptcy prediction, ARIMA is used to forecast the values of core financial indicators, such as total liabilities and total assets. If the predicted value of total liabilities is larger than that of total assets, the company is likely to fail within the given time period.

The ARIMA model is more complicated as compared with those using multiple linear regression. Moreover, it requires extensive calculation and analytical operations. However, this drawback is compensated with enhanced precision of forecasts.

## 5. Conclusion

Conventional models of bankruptcy prediction are diverse, but most of them use either of the two analytical methods – multiple linear regression (Altman’s and Ohlson’s models) or time series forecasting (ARIMA model). Multiple linear regression models are more convenient as they derive a single response variable, by the value of which the probability of bankruptcy can be assessed, from a standard set of independent variables. The use of time series forecasting models requires a careful choice of forecasted variables and additional calculations. Meanwhile, time series forecasting models are more adaptable and can be used for a variety of tasks.

The introduction of more advanced bankruptcy prediction methods, such as neural networks, has not completely eliminated the use of conventional models for research and analytical purposes. Conventional bankruptcy forecasting models combine clearness, simplicity and a rather high degree of accuracy, which makes them irreplaceable tools for fast estimation of a company’s financial prospects. Meanwhile, not all conventional models yield the same accuracy of results for a particular data set. Thus, the most appropriate one needs to be selected according to the character of data available and the relative importance of variables.

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