When using Bayesian probability models to make financial forecasts, you don’t need to know much about probability theory. Bayesian methods can help you improve probability estimates using intuitive procedures.

Any subject based on mathematics can reach a complex depth, but it doesn’t have to be so.

## how to use

The way in which Bayesian probability is used in American companies depends on a certain degree of belief, not on the historical frequency of the same or similar events. However, the model is versatile. You can incorporate frequency-based beliefs into the model.

The following uses the rules and assertions of the school of thought in Bayesian probability that are related to frequency rather than subjectivity. The measurement of quantified knowledge is based on historical data. This view is particularly useful in financial modeling.

## About Bayes’ Theorem

The specific formula of Bayesian probability that we will use is called Bayes’ theorem, sometimes also called Bayes’ formula or Bayes rule. This rule is most commonly used to calculate the so-called posterior probability. The posterior probability is the conditional probability of an uncertain event in the future, which is based on relevant evidence related to it in history.

In other words, if you obtain new information or evidence and need to update the probability of an event, you can use Bayes’ Theorem to estimate this new probability.

The formula is:

P (A ∣ B) = P (A ∩ B) P (B) = P (A) × P (B ∣ A) P (B) where: P (A) = the probability of occurrence of A, called the prior probability P (A ∣ B) = the conditional probability of A given B will happen P (B ∣ A) = B the conditional probability given A will happen P (B) = the probability of B happening begin(aligned) &P (A | B) = frac{ P (A cap B) }{ P (B)} = frac{ P (A) times P (B | A) }{ P (B)} \ &textbf{where:} &P(A) = text{The probability of occurrence of A, called} \ &text{prior probability} \ &P(A|B) = text{conditional probability of A} \ &text{ B appears} \ &P(B|A) = text{Conditional probability of given B} \ &text{A appears} \ &P(B) = text{Probability of B appears} \ end {Align}

phosphorus(One∣Second)=phosphorus(Second)phosphorus(One∩Second)=phosphorus(Second)phosphorus(One)Xphosphorus(Second∣One)Where:phosphorus(One)=The probability of occurrence of A is calledPriori probabilityphosphorus(One∣Second)=The conditional probability of A is givenB happensphosphorus(Second∣One)=Conditional probability of given BA happensphosphorus(Second)=Probability of occurrence of B

P(A|B) is the posterior probability because it depends on B. This assumes that A is not independent of B.

If we are interested in the probability of an event for which we have a priori observation, we call it a priori probability. We will consider this event A, and its probability P(A). If there is a second event that affects P(A), we call it event B, then we want to know the probability of A given B occurs.

In probability notation, this is P(A|B), which is called posterior probability or modified probability. This is because it happened after the original event, so it is postponed.

This is how Bayes’ theorem uniquely allows us to update our previous beliefs with new information. The following example will help you understand how it works in concepts related to the stock market.

## one example

Suppose we want to know how changes in interest rates will affect the value of stock market indexes.

All major stock market indexes have a lot of historical data available, so you should be able to find the results of these events easily. In our example, we will use the following data to understand the response of the stock market index to rising interest rates.

here:

P(SI) = probability of stock index rising

P(SD) = probability of stock index falling

P(ID) = probability of interest rate falling

P(II) = probability of rising interest rates

So the equation will be:

P (SD ∣ II) = P (SD) × P (II ∣ SD) P (II) begin(aligned) &P (SD | II) = frac{ P (SD) times P (II | SD)} {P (II)} \ end{aligned}

phosphorus(secondD∣A generationA generation)=phosphorus(A generationA generation)phosphorus(secondD)Xphosphorus(A generationA generation∣secondD)

Inserting our numbers, we get the following result:

P (SD ∣ II) = (1, 1 5 0 2, 0 0 0) × (9 5 0 1, 1 5 0) (1, 0 0 0 2, 0 0 0) = 0. 5 7 5 × 0. 8 2 6 0. 5 = 0. 4 7 4 9 5 0. 5 = 0. 9 4 9 9 ≈ 9 5% begin{aligned} P (SD | II) &= frac{ left (frac{ 1,150 }{ 2,000} right) times left (frac {950 }{ 1,150 } right) }{ left (frac {1,000 }{ 2,000} right)} \ &= frac{ 0.575 times 0.826 }{ 0.5} \ &= frac{ 0.47495 }{ 0.5} \ &= 0.9499 approx 95%\ end{aligned}

phosphorus(secondD∣A generationA generation)=(2,1,)(2,1,15)X(1,1595)=.5.575X.826=.5.47495=.9499≈95%

The table shows that the stock index dropped 1,150 out of 2,000 observations. This is based on the prior probability of historical data, in this case 57.5% (1,150/2,000).

This probability does not consider any information about interest rates, which we hope to update. After updating this prior probability with the information about the increase in interest rates, we update the probability of the stock market falling from 57.5% to 95%. Therefore, 95% is the posterior probability.

## Modeling with Bayes’ theorem

As mentioned above, we can use the results of historical data to build beliefs that we use to derive new update probabilities.

This example can be extrapolated to individual companies by using changes in their own balance sheets, bonds with changing credit ratings, and many other examples.

So, what if people don’t know the exact probability and only have estimates? This is where subjective opinions play an important role.

Many people attach great importance to estimates and simplified probabilities given by experts in their field. This also allows us to confidently generate new estimates for the new and more complex issues that are unavoidable obstacles in financial forecasting.

If we start with the right information, we can now use Bayes’ theorem instead of guessing.

## When to apply Bayes’ theorem

Changes in interest rates can greatly affect the value of specific assets. Therefore, changes in the value of assets will greatly affect the value of specific profitability and efficiency ratios used to represent company performance. The estimated probability is widely found to be related to the systematic changes in interest rates, so it can be effectively used in Bayes’ theorem.

We can also apply this process to the company’s net income stream. Litigation, changes in raw material prices, and many other things can affect the company’s net income.

By using probability estimates related to these factors, we can apply Bayes’ Theorem to find out what is important to us. Once we have found the derived probabilities we are looking for, we can simply apply mathematical expectations and outcome predictions to quantify financial probabilities.

Using countless related probabilities, we can use a simple formula to derive the answer to a fairly complex question. These methods are widely accepted and have passed the test of time. If applied properly, their use in financial modeling can help.

.