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Risk modeling – Strategy @ Risk

Strategy @ Risk

Tag: Risk modeling

The fallacies of Scenario analysis
This entry is part 1 of 4 in the series The fallacies of scenario analysis

Scenario analysis is often used in company valuation – with high, low and most likely scenarios to estimate the value range and expected value. A common definition seems to be:

Scenario analysis is a process of analyzing possible future events or series of actions by considering alternative possible outcomes (scenarios). The analysis is designed to allow improved decision-making by allowing consideration of outcomes and their implications.

Actually this definition covers at least two different types of analysis:
1. Alternative scenario analysis; in politics or geo-politics, scenario analysis involves modeling the possible alternative paths of a social or political environment and possibly diplomatic and war risks – “rehearsing the future”,
2. Scenario analysis; a number of versions of the underlying mathematical problem are created to model the uncertain factors in the analysis.
The first addresses “wicked” problems; ill-defined, ambiguous and associated with strong moral, political and professional issues. Since they are strongly stakeholder dependent, there is often little consensus about what the problem is, let alone how to resolve it. (Rittel & Webber,1974)

The second cover “tame” problems; that has well-defined and stable problem statements and belongs to a class of similar problems which are all solved in the same similar way. (Conklin, 2001) Tame however does not mean simple – a tame problem can be very technically complex.

Scenario analysis in the last sense is a compromise between computational complex stochastic models (the S&R approach) and the overly simplistic and often unrealistic deterministic models. Each scenario is a limited representation of the uncertain elements and one sub-problem is generated for each scenario.

Best Case/ Worse Case Scenarios analysis.
With risky assets, the actual cash flows can be very different from expectations. At the minimum, we can estimate the cash flows if everything works to perfection – a best case scenario – and if nothing does – a worst case scenario.

In practice, each input into asset value is set to its best (or worst) possible outcome and the cash flows estimated with those values.

Thus, when valuing a firm, the revenue growth rate and operating margin etc. is set at the highest possible level while interest rates etc. is set at its lowest level, and then the best-case scenario value is computed.

The question now is – if this really is the best (or worst) value or if let’s say a 95% (5%) percentile is chosen for each input – will that give the 95% (5%) percentile for the firm’s value?

Let’ say that we in the first case – (X + Y) – want to calculate entity value by adding ‘NPV of market value of FCF’ (X) and ‘NPV of continuing value’ (Y). Both are stochastic variables, X is positive while Y can be positive or negative. In the second case – (X – Y) – we want to calculate the value of equity by subtracting value of debt (Y) from entity value (X). Both X and Y are stochastic, positive variables.

From statistics we know that for the joint distribution of (X ±Y) the expected value E(X ±Y) is E(X) ± E(Y) and that Var(X ± Y) is Var(X) + Var(Y) ± 2Cov(X,Y). Already from the expression for the joint variance we can see that this not necessarily will be true. However the expected value will be the same.

We can demonstrate this by calculating a number of percentiles for two normal independent distributions (with Cov(X,Y)=0, to make it simple) and add (subtract) them and plot the result (red line) with the same percentiles from the joint distribution – blue line for (X+Y) and green line for (X-Y).

As we can see the lines for X+Y only coincides at the expected value and the deviation increases as we move out on the tails. For X-Y the deviation is even more pronounced:

Plotting the deviation from the joint distribution as percentage from X Y, demonstrates very large relative deviations as we move out on the tails and that the sign of the numerical operator totally changes the direction of the deviations:

Add to this, a valuation analysis with a large number of:
1. both correlated and auto-correlated stochastic variables,
2. complex calculations,
3. simultaneous equations,
and there is no way of finding out where you are on the probability distribution – unless you do a complete Monte Carlo simulation. It is like being out in the woods at night without a map and compass – you know you are in the woods but not where.

Some advocates scenario analysis to measure risk on an asset using the difference between the best-case and worst-case. Based on the above this can only be a very bad idea, since risk in the sense of loss is connected to the left tail where the deviation from the joint distribution can be expected to be the largest. This brings us to the next post in the series.

References

Rittel, H., and Webber, M. (1973). Dilemmas in a General Theory of Planning. Policy Sciences, Vol. 4, pp 155-169. Elsevier Scientific Publishing Company, Inc: Amsterdam.

Conklin, Jeff (2001). Wicked Problems. Retrieved April 28, 2009, from CofNexus Institute Web site: http://www.cognexus.org/wpf/wickedproblems.pdf
04/05/2009
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
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