Financial Forecasting: The Bayesian Method

By Daniel McNulty AAA

If you don't know a lot about probability theory, Bayesian methods probably sounds like a scary topic. It's not. While any mathematically based topic can be taken to rather complex depths, the use of a basic Bayesian probability model in financial forecasting can help refine probability estimates using an intuitive process.

Bayesian Probability
Bayesian probability's application in corporate America is highly dependent on the "degree of belief" rather than historical frequencies of identical or similar events. You can also use your historical beliefs based on frequency to use the model; it's a very versatile model.

For this article, we will be using the rules and assertions of the school of thought that pertains to frequency rather than subjectivity within Bayesian probability. This means that the measurement of knowledge that is being quantified is based on historical data. This view of the model is where it becomes particularly helpful in financial modeling. The application of how we can integrate this into our models is explained in the section to follow.

Bayes' Theorem
The particular formula from Bayesian probability we are going to use is called Bayes' Theorem, sometimes called Bayes' formula or Bayes' rule. This particular rule is most often used to calculate what is called the posterior probability. The posterior probability is the conditional probability of a future uncertain event that is based upon relevant evidence relating to it historically. In other words, if you gain new information or evidence and you need to update the probability of an event occurring, you can use Baye's Theorem to estimate this new probability.
The formula is:


P(A) is the probability of A occurring, and is called the prior probability.
P(A|B) is the conditional probability of A given that B occurs. This is the posterior probability due to its variable dependency on B. This assumes that the A is not independent of B.

P(B|A) is the conditional probability of B given that A occurs.

P(B) is the probability of B occurring.

If we are interested in the probability of an event of which we have prior observations; we call this the prior probability. We'll deem this event event A, and its probability P(A). If there is a second event that affects P(A), which we'll call event B, then we want to know what the probability of A is given B has occurred. In probabilistic notation this is P(A|B), and is known as posterior probability or revised probability. This is because it has occurred after original event, hence the post in posterior. This is how Bayes' theorem uniquely allows us to update our previous beliefs with new information. The example below will help you see how it works while incorporating it within an equity market concept.

An Example
Let's say we want to know how a change in interest rates would affect the value of a stock market index. All major stock market indexes have a plethora of historical data available so you should have no problem finding the outcomes for these events with a little bit of research. For our example we will use the data below to find out how a stock market index will react to a rise in interest rates.


Stock Price
Interest Rates
Decline Increase Unit Frequency
Decline 200 950 1150
Increase 800 50 850
1000 1000 2000

Here:

P(SI) = the probability of the stock index increasing
P(SD) = the probability of the stock index decreasing
P(ID) = the probability of interest rates decreasing
P(II) = the probability of interest rates increasing

So the equation will be:


Thus with our example plugging in our number we get:


In the table you can see that out of 2000 observations, 1150 instances showed the stock index decreased. This is the prior probability based on historical data, which in this example is 57.5% (1150/2000). This probability doesn't take into account any information about interest rates, and is the one we wish to update. After updating this prior probability with information that interest rates have risen leads us to update the probability of the stock market decreasing from 57.5% to 95%. 95% is the posterior probability.

Modeling with Bayes' Theorem
As seen above we can use the outcomes of historical data to base our beliefs on from which we can derive new updated probabilities. This example can be extrapolated to individual companies given changes within their own balance sheets, bonds given changes in credit rating, and many other examples. (Learn how to analyze the balance sheet in our article, Breaking Down The Balance Sheet.)

So what if one does not know the exact probabilities but has only estimates? This is where the subjectivists' view comes strongly into play. Many people put a lot of faith into the estimates and simplified probabilities given by experts in their field; this also gives us the great ability to confidently produce new estimates for new and more complicated questions introduced by those inevitable roadblocks in financial forecasting. Instead of guessing or using simple probability trees to overcome these road blocks, we can now use Bayes' Theorem if we possess the right information with which to start. (See Analyst Forecasts Spell Disaster For Some Stocks to read about the effects of a bad forecast.)

Now that we have learned how to correctly compute Bayes' Theorem, we can now learn just where it can be applied in financial modeling. Other, and much more inherently complicated business specific, full-scale examples will not be provided, but situations of where and how to use Bayes' Theorem will.

Changing interest rates can heavily affect the value of particular assets. The changing value of assets can therefore greatly affect the value of particular profitability and efficiency ratios used to proxy a company's performance. Estimated probabilities are widely found relating to systematic changes in interest rates and can therefore be used effectively in Bayes' Theorem.

Another avenue where we can apply our newfound process is in a company's net income stream. Lawsuits, changes in the prices of raw materials, and many other things can heavily influence the value of a company's net income. By using probability estimates relating to these factors, we can apply Bayes' Theorem to figure out what is important to us.

Once we find the deduced probabilities that we are looking for it is only a simple application of mathematical expectancy and result forecasting in order to monetarily quantify our probabilities.

Conclusion
To conclude, we found that by using a myriad of related probabilities we can deduce the answer to rather complex questions with one simple formula. These methods are well accepted and time tested, their use in financial modeling can be very helpful and advantageous if applied properly.

For further reading on another forecasting technique, take a look at Multivariate Models: The Monte Carlo Analysis.

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