Artem has a doctor of veterinary medicine degree.
What Is Risk?
In finance and economics, risk is a term that's related to uncertainty about an event and its outcome, regardless of whether the event and outcome are positive or negative. A good example of this is the risk of making a financial investment. We are uncertain about the outcome of investing in a stock, and may quantify our uncertainty of loss and/or gain via a probability distribution model.
However, some models of risk are quite subjective. This is because some of our assumptions about risk deal with a person's individual attitudes toward risk and their understanding of a specific situation.
Let's go over how this is the case as we explore the concept of risk attitudes and their intersection with Bayesian inference and probabilities.
Types of Attitudes
First, let's quickly define the basics of the different types of risk attitudes.
- Risk aversion is a type of attitude where an individual gravitates toward certain, as opposed to uncertain, events.
- Risk seeking is a type of attitude or behavior where a person is inclined to take on less-certain activities in lieu of more certain ones.
- In the middle are risk neutral individuals, who have an indifferent attitude toward risk.
A Mathematical Look
Let's go over those concepts in a more mathematical manner.
Say you own a business where you can either make $100,000 or $200,000 this year, with an inherent probability of either being 0.5. Thus, the expected value of this scenario is right in the middle: $150,000.
A competitor comes to you and offers to buy your business for $140,000, guaranteed. In other words, the probability of making $140,000 in this case is 1.
The problem is that we don't know the actual probability of making either $100,000 or $200,000 because we don't know the true probability of the parameter, the process, which actually leads to one or the other. So, we have to partially rely on a set of subjective assumptions to create a model that takes into account our attitudes about the situation. This is one of the core principles of Bayesian inference.
For instance, we could envision a scenario where our prior belief about our business is strongly skewed toward the attitude that the probability of earning $100,000 is far higher than the probability of earning $200,000. This would mean our assessment about the actual expected value would yield a value that is less than that of the inherent expected value.
Or, conversely, we might start out believing that the probability of either scenario is exactly that of the inherent probability. However, during negotiations, we come across a piece of data that significantly skews our attitude about the business's income this year. We come to believe that the probability of making $100,000 is now far greater than it was before this new information.
Either way, our interpretation yields a subjective expected value that is less than the inherent expected value. Because of this, we take the offer of a guaranteed $140,000 payoff because we believe the actual payoff will actually be far less than that.
A person is said to exhibit a risk-averse attitude when their interpretation of the expected value is less than the actual expected value. This means they leave a good chance of higher profits on the table, look for a more certain scenario instead, and settle for less than would be expected as a result.
Now let's go over the flip-side. The scenario is exactly as before, except the competitor ups the price they're willing to pay to $150,000 to try and sweeten the deal.
However, this time we have a prior belief that tells us that the probability of making $200,000 this year is far greater than 0.5. Prior beliefs like this can be based on purely subjective attitudes based on personal experience. In this case, that could be nothing more than a gut feeling.
Or, such prior beliefs that determine these probabilities can be based on data. For instance, you might understand the true income trends of your business and come to believe that the probability of making $200,000 is quite high. Yet it's also possible that you really do believe that the probability of making $200,000 is equal to that of the $100,000.
However, as negotiations over the selling price move forward, a new piece of information comes about that changes your beliefs. For instance, maybe a client of yours calls and makes a verbal agreement to buy more from you. This makes you believe you'll make more this year than you did before.
Again, either way, the resultant (posterior) probability of making $200,000 is now far higher in your view. This ultimately leads to an expected value that is also far higher than $150,000.
In this case, you turn down the offer of $150,000 because your attitude about the expected value is higher than the true expected value of $150,000. Meaning, you believe you'll make more than $150,000 and are willing to risk forgoing a 100% certain payoff for potentially more (at the risk of making much less if you only make $100,000).
In cases where a person's attitude about the expected value is such that their expected value is higher than the actual expected value, the person is said to be risk-seeking.
As you can expect, if your selling price is equal to that of the expected value, then you are at a risk-neutral price and will be happy to sell your business for exactly $150,000.
From a financial or economics perspective, risk refers to uncertainty, but not necessarily something good or bad. People with a risk seeking attitude gravitate toward ventures where their personal value (judgment) of the event and its outcome is higher than the mathematical expected value. In contrast, individuals who are risk averse are happy moving toward more certain events and/or those where their valuation of the event and outcome is lower than the mathematical expected value. A risk neutral individual is indifferent about uncertainty and is willing to part with an opportunity for the expected value of the opportunity.
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