Ivan Bercovich Blogs

This is the second time I attempt to write a post about this topic. Initially, I was focusing too much on how different cultures see failure. But realizing my lack of knowledge on the topic, I decided to abstain—I would not like to fall into false stereotypes or offend anybody. However, I am really interested in how different cultures perceive failure—particularly how this perception differs between western and eastern cultures. So if anybody would like to share some personal stories related to the topic, I would be very happy.

Now to the point.

Below, you can see a Gaussian distribution, which you might also know as a bell curve. Since probability and statistics is a field in which I feel very comfortable, I am going to wrap this discussion around it. But first some notes.

The Gaussian:

The Gaussian distribution has been horribly misused by a lot of people for a long time (read: Recipe for Disaster: The Formula That Killed Wall Street). So please, don’t assume what I am about to describe can actually be characterized using a Gaussian distribution. In practice, probability distributions and mathematical models that try to describe how a socioeconomic process works are incredibly complex (and almost always incorrect). However, this distribution is a good tool to explain how how probability affects our everyday lives.

Risk (finance) or Variance (math):

In practice, it is very hard (rather impossible) to come up with a numerical value to measure the variance of complex processes, such as the ones involved in the financial markets. However, in conjunction with the Gaussian distribution, it allows us to envision the real world with a simple probabilistic model.

Now that the weak points of what follows have been covered, for simplicity’s sake, I am going to continue writing as if the Gaussian was in fact a perfect representation of the world.

So here is how the model works. For every process that we can image (which has a large sample space*), there is a perfectly symmetric Gaussian distribution, with a known mean and variance. The mean is the value that lives at the center of the distribution (x-axis point of symmetry). The variance determines the width of the distribution, and therefore, the likelihood of an extreme event happening. We will call this extreme events “outliers”.

As you can see in the picture above, the larger the variance (represented as sigma squared), the wider the distribution. Note how the width grows equally in both ends of the distribution, meaning the distribution always stays symmetric. We will call the right side of the distribution “the upside” and the left side “the downside”.

It should be clear by now that in order to have a really high upside, we will have to accept the possibility of a really bad downside as well. Given this model, we have two ways of running our lives. One would be to minimize risk (again, I am assuming we know what those risks are), we don’t go climbing, we don’t take too many courses, we don’t eat too much red meat, we don’t travel, etc. In this manner, we are guaranteeing ourselves the mean. We will most likely achieve an average result in everything we do, and it will be very unlikely we will ever fail.

The other option is to take a lot of risks. We would travel the world for a year, take 8 classes a semester, learn how to fly planes, and go to a concert the day before a final exam. This strategy will widen the range of possible results, but we won’t get to pick between upside and downside results. We might meet Richard Branson in our trip and start a profitable relationship with him, or the plane we are learning how to fly might fall on our first attempt.

Of course these situations are extremes, but they do a good job at illustrating the point. But how can we exploit this duality and benefit from it? I will provide my own version of this answer in my next post.

Recommended Books:

• The Black Swan
• Super Crunchers
• Probability and Stochastic Processes

 

* A large sample space means that the range of possibilities is not a small number. For example, in a test I can get a score between 0 and 10 and the material that will go into the exam is also very defined. So the more I study, the better I will perform—this is not the kind of process being discussed here. But if the grading scale was very large or if the material was more than anybody can study, then it would become more of a random process.