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Risk of Bankruptcy – Strategy @ Risk

Strategy @ Risk

Series: Risk of Bankruptcy

The Risk of Bankruptcy

This entry is part 1 of 4 in the series Risk of Bankruptcy

Investors should be skeptical of history-based models. Constructed by a nerdy-sounding priesthood using esoteric terms such as beta, gamma, sigma and the like, these models tend to look impressive. Too often, though, investors forget to examine the assumptions behind the symbols. Our advice: Beware of geeks bearing formulas. – Warren E. Buffett. ((Buffett, Warren E., “Shareholder Letters.” Berkshire Hathaway Inc. 27 February 2009,. Berkshire Hathaway Inc. 13 Mar 2009 <http://www.berkshirehathaway.com/letters/letters.html>.))

Historic growth is usually a risky estimate for future growth. To be able to forecast a company’s future performance you have to make assumptions on the future most likely values and their event space of a large number of variables, and then calculate both the probability of future necessary cash infusions and if they do not materialized – the risk of bankruptcy.

The following calculations are carried out using the Strategy& Risk simulation model. Such simulations can be carried out on all types of enterprises including the financial sector. There are several models in use for predicting bankruptcy and we have in our balance simulation model implemented two; Altman’s Z-score model and the risk index Z developed by Hannan and Hanweck.

Atman’s Z-score model is based on financial ratios and their relation to bankruptcy found from discriminant analysis. ((Altman, E. I.. “Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy.” The Journal of Finance 23(1968): 589-609. )) The coefficients in the discriminant function has in later studies been revised – the Z’-score and Z’’-score models.

Hannan and Hanweck’s probability of insolvency is based on the likelihood of return to assets being negative and larger then the capital-asset ratio. ((Timothy H., Hannan, Gerald A. Hanweck. “Bank Insolvency Risk and the Market for Large Certificates of Deposit.” Journal of Money, Credit and Banking 20(1988): 203-211.)) The Z index has been used to forecast bank insolvency ((Kimball, Ralph C.. “Economic Profit and Performance Measurement in Banking.” New England Economic Review July/August(1998): 35-53.)) ((Jordan, John S.. “Problem Loans at New England Banks, 1989 to 1992: Evidence of Aggressive Loan Policies.” New England Economic Review January/February(1998): 23-38.)), but can profitably be used to study large private companies with low return to assets.

We will here take a look at the Z-index and in a later post use the same data to calculate the Z-scores.

The following calculations are based on forecasts, EBITDA and balance simulations – not on historic balance sheet data. The Z-index is defined as:

Z=

where ROA is the pre-tax return on assets, K the ratio of equity to assets, and s the standard deviation of pre-tax ROA. The Z-index give pr unit of standard deviation of ROA the decline in ROA the company can manage before equity is exhausted and becomes insolvent.

We will in the simulation (250 runs) for every year in the 15 year forecast period – both forecast the yearly ROA and K, and use the variance in ROA to estimate s. For every value of Z – assuming a symmetric distribution – we can calculate the perceived probability (upper bound) of insolvency (p) from:

P =

where the multiplication by (1/2) reflects the fact that insolvensy occurs only in the left tail of the distribution. The relation of p to Z is inverse one, with higher Z-ratios indicating low probability of insolvency.

Since our simulation cover a 15 year period it is fully possible that multi-period losses, thru decline in K, can wipe out the equity and cause a failure of the company.

In year five of the simulation the situation is as follows, the pre-tax return on assets is low – on average only 1.3% and in 20% of the cases it is zero or negative.

However the ratio of equity to assets is high – on average 37% with standard deviation of only 1.2.

The distribution of the corresponding Z-Index values is given in the chart below. It is skewed with a long right tail; the mean is 32 with a minimum value of 16.

From the graph giving the relation between the Z-index and probability of insolvency it is clear that the company’s economic situation is far from being threatened. If we look at the distribution for the probability of insolvency as calculated from the estimated Z-index values this is confirmed having values in the range from 0.1 to 0.3.

Having the probability of insolvency pr year gives us the opportunity to calculate the probability of failure over the forecast period for any chosen strategy.

If it can’t be expressed in figures, it is not science; it is opinion. It has long been known that one horse can run faster than another — but which one? Differences are crucial. ((Heinlein, Robert. Time Enough for Love. New York: Putnam, 1973))

References

16/03/2009
Predicting Bankruptcy
This entry is part 2 of 4 in the series Risk of Bankruptcy

The Z-score formula for predicting bankruptcy was developed in 1968 by Edward I. Altman. The Z-score is not intended to predict when a firm will file a formal declaration of bankruptcy in a district court. It is instead a measure of how closely a firm resembles other firms that have filed for bankruptcy.

The Z-score is classification method using a multivariate discriminant function that measures corporate financial distress and predicts the likelihood of bankruptcy within two years. ((Altman, Edward I., “Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy”. Journal of Finance, (September 1968): pp. 589-609.))

Others like Springate ((Springate, Gordon L.V., “Predicting the Possibility of Failure in a Canadian Firm”. Unpublished M.B.A. Research Project, Simon Fraser University, January 1978.)), Fulmer ((Fulmer, John G. Jr., Moon, James E., Gavin, Thomas A., Erwin, Michael J., “A Bankruptcy Classification Model For Small Firms”. Journal of Commercial Bank Lending (July 1984): pp. 25-37.)) and the CA-SCORE model ((“C.A. – Score, A Warning System for Small Business Failures”, Bilanas (June 1987): pp. 29-31.)) have later followed in Altman’s track using step-wise multiple discriminant analysis to evaluate a large number of financial ratio’s ability to discriminate between corporate future failures and successes.

Since Altman’s discriminant function only is linear in the explanatory variables, there has been a number of attempts to capture non-linear relations thru other types of models ((Berg, Daniel. “Bankruptcy Prediction by Generalized Additive Models.” Statistical Research Report. January 2005. Dept. of Math. University of Oslo. 20 Mar 2009 <http://www.math.uio.no/eprint/stat_report/2005/01-05.pdf>.)) ((Dakovic, Rada,Claudia Czado,Daniel Berg. Bankruptcy prediction in Norway: a comparison study. June 2007. Dept. of Math. University of Oslo. 20 Mar 2009 <http://www.math.uio.no/eprint/stat_report/2007/04-07.pdf>.)). Even if some of these models shows a somewhat better predicting ability, we will use the better known Z-score model in the following.

Studies measuring the effectiveness of the Z-score claims the model to be accurate with >70% reliability. Altman found that about 95% of the bankrupt firms were correctly classified as bankrupt. And roughly 80% of the sick, non-bankrupt firms were correctly classified as non-bankrupt (( Altman, Edward I.. “Revisiting Credit Scoring Models in a Basel 2 Environment.” Finance Working Paper Series . May 2002. Stern School of Business. 20 Mar 2009 <http://w4.stern.nyu.edu/finance/docs/WP/2002/html/wpa02041.html>. )). However others find that the Z-score tends to misclasifie the non-bankrupt firms ((Ricci, Cecilia Wagner. “Bankruptcy Prediction: The Case of the CLECS.” Mid-American Journal of Business 18(2003): 71-81.)).

The Z-score combines four or five common business ratios using a linear discriminant function to determine the regions with high likelihood of bankruptcy. The discriminant coefficients (ratio value weights) were originally based on data from publicly held manufacturers, but have since been modified for private manufacturing, non-manufacturing and service companies.

The original data sample consisted of 66 firms, half of which had filed for bankruptcy under Chapter 7. All businesses in the database were manufacturers and small firms with assets of <$1million was eliminated.

The advantage of discriminant analysis is that many characteristics can be combined into a single score. A low score implies membership in one group, a high score implies membership in the other group, and a middling score causes uncertainty as to which group the subject belongs.

The original score was as follows:

where:

= Working Capital / Total Assets, = Retained Earnings / Total Assets
= EBIT/ Total Assets, = Sales/ Total Assets
= Market Value of Equity / Book Value of Total Liabilities

From about 1985 onwards, the Z-scores have gained acceptance by auditors, management accountants, courts, and database systems used for loan evaluation. It has been used in a variety of contexts and countries, but was designed originally for publicly held manufacturing companies with assets of more than $1 million. Later revisions take into account the book value of privately held shares, and the fact that turnover ratios vary widely in non-manufacturing industries:
1. Z-score for publicly held Manufacturers
2. Z’-score for private Firms
3. Z’’-score for Manufacturers, Non-Manufacturer Industrials & Emerging Market Credits
The estimated discriminant coefficients for the different models is given in the following table: [Table=3] and the accompanying borders of the different regions – risk zones – are given in the table below. [Table=4] In the following calculations we will use the estimated value of equity as a proxy for market capitalization. Actually it is the other way around since the market capitalization is a guesstimate of the intrinsic equity value.

In our calculations the Z-score metrics will become stochastic variables with distributions derived both from the operational input distributions for sale, prices, costs etc. and the distributions for the financial variables like risk free interest rate, inflation etc. The figures below are taken from the fifth year in the simulation to be comparable with the previous Z-index calculation that gave a very low probability for insolvency.

We have in the following calculated all three Z metrics, even when only the Z-score fits the company description.

Using the Z-score metric we find that the company with high probability will be found in the distress area – it can even have negative Z-score. The last is due to the fact that the company has negative working capital – being partly financed by its suppliers and partly to the use of calculated value of equity – which can be negative.

The Z’’-score is even more somber giving no possibility for values outside the distress area:

The Z’’-score however puts most of the observations in the gray area:

Before drawing any conclusions we will in the next post look at the time series for both the Z-index and the Z-scores. Nevertheless one observation can be made – the Z metric is a stochastic variable with an event space that easily can encompass all three risk zones – we therefore need the probability distribution over the zones to forecast the risk of bankruptcy.

References
20/03/2009
The Probability of Bankruptcy

This entry is part 3 of 4 in the series Risk of Bankruptcy

In the simulation we have for every year calculated all four metrics, and over the 250 runs their mean and standard deviation. All metrics is thus based on the same data set. During the forecast period the company invested heavily, financed partly by equity and partly by loans. The operations admittedly give a low but fairly stable return to assets. It was however never at any time in need for capital infusion to avoid insolvency. Since we now “know” the future we can judge the metrics ability to predict bankruptcy.

A good metric should have a low probability of rejecting a true hypothesis of bankruptcy (false positive) and a high probability of rejecting a false hypothesis of bankruptcy (false negative).

In the figures below the more or less horizontal curve gives the most likely value of the metric, while the vertical red lines indicate the 90% event space. By visual inspection of the area covered by the red lines we can get an indication of the false negative and false positive rate.

The Z-Index shows an increase over time in the probability of insolvency, but the probability is very low for all years in the forecast period. The most striking effect is the increase in variance as we move towards the end of the simulated period. This is caused by the fact that uncertainty is “accumulated” over the forecast period. However, according to the Z-index, this company will not be endangered inside the 15 year horizon.

In our case the Z-Index correctly identifies the probability of insolvency as small. By inspecting the yearly outcomes represented by the vertical lines we also find an almost zero false negative rate.

The Z-score metrics tells a different story. The Z’’-score starts in the grey area and eventually ends up in the distress zone. The two others put the company in the distress zone for the whole forecast period.

Since the distress zone for the Z-score is below 1.8, a visual inspection of the area covered by the red lines indicates that most of the outcomes fall in the distress zone. The Z-score metrics in this case performs type II errors by giving false negative judgements. However it is not clear what this means – only that the company in some respect is similar to companies gone bankrupt.

If we look at the Z metrics for the individual years we find that the Z-score have values from minus two to plus three, in fact it has a coefficient of variation ranging from 300% to 500%. In addition there is very little evidence of the expected cumulative effect.

The other two metrics (Z’ and Z’’) shows much less variation and the expected cumulative effect. The Z’-score outcomes fall entirely in the distress zone, giving a 100% false negative rate.

The Z’’-score outcome falls mostly in the distress zone below 1.1, but more and more falls in the grey area as we move forward in time. If we combine the safe zone with the grey we get a much lower false negative rate than for both the Z and the Z’ score.

It is difficult to draw conclusions from this exercise, but it points to the possibility of high false negative rates for the Z metrics. Use of ratios in assessing a company’s performance is often questionable and a linear metric based on a few such ratios will obviously have limitations. The fact that the original sample consisted of the same number of healthy and bankrupt companies might also have contributed to a bias in the discriminant coefficients. In real life the failure rate is much lower than 50%!

30/03/2009
Credit Risk
This entry is part 4 of 4 in the series Risk of Bankruptcy

Other Methods

A number of other statistical methods have also been used to predict future company failure and credit risk, see: (Atiya, 2001), (Chandra, Ravi, Bose, 2009) and (Bastos, 2008). A recent study (Boguslauskas, Mileris , 2009) analyzed 30 scientific publications comprising 77 models:
1. 63% used artificial neural networks (ANN)
2. 53% used logistic regression (LR)
3. 37% used discriminant analysis (DA)
4. 23% used decision trees and (DT)
5. 33% used various other methods
The general accuracy of the different models was evaluated: the proportion of companies correctly classified (figure 3 in the article):

The box and whisker plot above shows that logistic regression (87%) and artificial neural networks (87%) gives almost the same accuracy while decision trees (83%) and discriminant analysis (77%) seems to be less reliable methods.

However from the boxes it is evident that decision trees as a method have a much larger variance in classification accuracy than the others and that artificial neural network have the lowest variance. For logistic regression and discriminant analysis the variance is approximately the same.

Comparing methods based on different data sets can easily be misleading. Accurate parameter estimation relies heavily on available data and their usability for that particular method.

References

Atiya, Amir F. (2001). Bankruptcy prediction for credit risk using neural networks: a survey and new results. IEEE TRANSACTIONS ON NEURAL NETWORKS, 12(4), Retrieved from http://ieee-cis.org/pubs/tnn/

Bastos, Joao. (2008, April 01). Credit scoring with boosted decision trees. Retrieved from http://mpra.ub.uni-muenchen.de/8156/

Boguslauskas, Vytautas. Mileris, Ricardas. (2009). Estimation of credit risk by artificial neural networks models. ECONOMICS OF ENGINEERING DECISIONS, 4(64), Retrieved from http://internet.ktu.lt/en/science/journals/econo/inzek064.html

Chandra, D. K., Ravi, V., Bose, I. (2009). Failure prediction of dotcom companies using hybrid intelligent techniques. Expert Systems with Applications, (36), 4830–4837.
29/12/2009