Survival Rate of a Company — the Gambler’s Ruin
A structural underestimation of how frequent, really bad and highly contagious things can happen in a market, the possibility of a swift build up of even small risk positions into heavy risk clusters, or simply the reliance on wrong risk assumptions will inevitably lead to a toxic cocktail which might force a company to exit its business rather sooner than later.
It is amazing how fast this forced exit can come forward and how it gets market participants by surprise.
In this article, we have a look at the speed of events unraveling and how all this may shorten the lifetime of a company. To put it another way, we have a look at this problem from a Gambler’s Ruin perspective.
An Extremely Unlikely Threat
Let’s assume that a company has the following profit/ loss expectation within each period of time (e.g. each quarter):
The expected profit of the company per time period is EUR 3 millions on average (the black dotted line in the graph above). Though there is the chance of achieving higher profits, there is also the possibility of making a loss. In fact, the probability of making a loss is around 35% (dark grey area in the graph).
As a serious constraint, our company will get busted in case there is a loss of -EUR 20 millions (the red dotted line in the graph above) or beyond accruing in any given period of time. In case this happens, the company exits its business. Hence, there will be no more operations afterwards, the company ceases to exist — a Gambler’s Ruin.
Nevertheless, the chances for such a scenario are incredibly small with a probability of just 0.2%. This is an unbelievably negligible risk factor one would be tempted to say.
This whole profit/ loss pattern is based on the assumption, that the earnings of the company in principle follow a thin tailed distribution (here: normal distribution). This means that the company does not expect profits or shortfalls too far away from the average (remember: average earnings are EUR 3 millions per time period).
In other words, it is extremely unlikely for really bad things to happen in the market (something like “1 in 10,000 days…”).
A More Accentuated Risk Behaviour
The profit and loss expectations for the same company look a bit different when we allow for a higher tail risk (i.e. bad events could affect the market more frequently):
Also here, the expected average profit is EUR 3 millions with a probability of loss in any given time period amounting to 36%. So far — no real changes occurred compared to the scenario from before.
Though, it is to be noted that the maximum loss per period can be much higher and the probability of hitting the deadly benchmark of -EUR 20m is already up to 2.3%.
This still might seem not too much a risk. But by simply changing the assumption of the market’s risk behaviour, the threat of being forced out of the market increased already by a factor of 12.
To be sure, here we incorporated just a moderate fat tailed distribution (here: Student’s t distribution with degrees of freedom = 4). Hence, bad market events with significant negative impact happen but do surface with a still moderate frequency.
The following graph gives an impression how the tail risk (= the risk of heavy loss) changes when moving from a thin tailed risk behaviour into a moderate fat tailed risk behaviour:
Factor Time
This is the part where it really gets nasty. We introduce time as a risk factor.
So far, the focus was on the profit/ loss pattern for any given period of time. And, for each time period the danger of getting destroyed by market circumstances was either insignificant (in case of the thin tailed distribution) or was at least still very low (in case of the moderate fat tailed distribution).
But business quarters come and go. And with this, even very small risk factors are starting to build up.
Here is a comparison between those two risk behaviours (thin tailed and moderate fat tailed) where the number of uneventful time periods play a role (i.e. time periods where no impactful negative market events happened):
The graph shows how long a company — without adjusting its risk position — will operate in a given market framework without being forced to exit.
In a market where the occurrence of crazy events is extremely unlikely (represented by the thin tailed distribution), the company may operate for quite a long time until it hits the bottom.
In a market framework where bad scenarios happen more frequently (represented by a moderate fat tailed distribution), the life span of the company is significantly shortened.
Accordingly, in the first case the chances of getting busted already in the first quarter is 0.2%. The probability for the second scenario is 2.3%. This is of course insignificant or at least quite low and corresponds to what we said in the previous chapter.
The chances of getting busted after 10 quarters (i.e. after 2.5 years) are up to 2.2% in the thin tailed market environment BUT increased already to 22% in the moderate fat tailed market environment. After 5 years the probabilities raised to 4% and 38% (!) respectively.
Imagine a company which thinks to operate in a thin tailed market environment but instead is actually embedded in a moderate fat tailed market framework. There will be an increased misconception of tolerated risk profile and risk profile in real terms.
As said in the intro, wrong market assumptions and the build up of rather small risk positions into a dominant risk cluster over time can provide some heavy surprises in a toxic momentum.
The following graph shows how fast this misperception of risk exposure grows over a relatively short period of time:
Survival Simulation
In order to get a feeling about the survival rate of a company in both of those market environments, we simulated those risk assumptions over a certain period of time.
In this simulation, a company operates e.g. quarter by quarter for a certain time period. As soon as the loss in a quarter hits -EUR 20 millions or beyond (exit benchmark represented by the red dotted line in the graphs below), the company has to exit its business — Gambler’s Ruin.
Here are some of the survival scenarios for the thin tailed market environment:
In general, the survival rate is pretty good. Though, even in this favourable market environment with a mere 0.2% chance of hitting the exit-button there are some short-lived exceptions.
And here, some scenarios for the moderate fat tailed environment:
The results are pretty clear. Without an adequate risk management, forced market exits are rather the rule than the exception.
No Extremes?
Let’s assume we are still in the thin tailed market environment.
Hence, extreme events on a more frequent basis are not expected to happen resp. are extremely unlikely.
Though, so far it was taken as granted that the scope of volatility in this market frame is stable. In other words, the underlying thin tailed probability distribution has a stable standard deviation (as a measure of volatility).
But what happens when this volatility framework is also becoming unstable? In this case, the standard deviation follows its own probability pattern.
The results in a simulation over a certain time frame are quite sobering:
With an unstable volatility umbrella, fat risk tails are immediately building up and as a consequence the number of companies being forced to exit the market is increasing significantly even in this thin tailed market environment.
It seems, there is no escape…
Conclusion
Now imagine, you think your company is operating in the first market scenario (thin tailed) while in real terms it is exposed to the second market scenario (moderate fat tailed) or it is operating even in a thin tailed environment with unstable volatility. Without any further countermeasures, this is like walking on very thin ice without even noticing it.
Wrong assumptions about the market behaviour and the underestimation of in fact small risk factors without taking the factor time into account can lead to a heavy misperception of the risk exposures a company is operating in.
And as the business world seldom matches thin tailed assumptions with stable frameworks, unpleasant surprises surface quite frequently.
Or as Nassim Nicholas Taleb puts it:
Every payoff one can think of in nature is nonlinear, hence subjected to some tail payoff, and some asymmetry in its distribution.
Statistical modelling, predictive analytics and a deep knowledge of the business are some of the more crucial tools to be incorporated in ordered to counteract those tendencies.
By the way, here we took a moderate fat tailed behaviour as an example. Needless to say that market patterns can be much, much more difficult. For a more detailed view on this, see:
Risk Management — Keeping Up Appearances | by Christian Schitton | Analytics Vidhya | Medium
References
A Map and Simple Heuristic to Detect Fragility, Antifragility, and Model Error by N.N. Taleb, New York University — Polytechnic Institute/ June 4, 2011
Risk Management — Keeping Up Appearances by Christian Schitton — Analytics Vidhya/ September 30, 2021
