# Bear Market Probability

The Bear Market Probability can be more technically described as the probability of a bear market given the yield curve ("P[BM|YC]"). Additionally, we define the Bear Market Probability as the chance of a Stock Market decline that is 20% or greater over the next 24 months. We also define the Yield Curve here as a comparison of the federal funds and 10 year treasury rates. Let's explore deeper on how the Bear Market Probability is calculated by examining the sources of data and describing the statistical computations.

## The Yield Curve

At the end of each month, and as far back as 1959, we collect the 10 year treasury rate from the U.S Treasury's website. During this same time, we acquire the end of month federal funds rate from the New York Federal Reserve's website. Dividing the federal funds rate by the 10 year treasury rate gives us a measurement that we call the Yield Curve and is used as the independent variable of our analysis of Stock Market return.

## The Stock Market Return

Using our Stock Price API, we collect S&P 500 monthly returns. From this data we calculate each monthly observation's minimum logarithmic return over the next 24 months of that observation. This is different from the normal measure of a return in that usual methods look backwards in time by taking the subject's value and comparing it to an observation in the past - whereas our calculation takes the subject's value and compares it to a future observation. We take this forward looking approach as we are interested in how the Stock Market behaves in the future given the interest rate environment at a given point in time. This result is used as the dependent variable within our analysis and we refer to it as the Stock Market Return.

## Building the Data Map

We round the data in an effort to consolidate it into measurable groups and for the ease of the analysis. The Yield Curve data is rounded to the nearest 0.05 and Stock Market Return data is rounded to the nearest 0.01. Using the Yield Curve rounded data as the X axis and the Stock Market Return rounded data as the Y axis, each time paired monthly observation is counted in the Data Map. This results in a table of counts, or histograms, of Stock Market Return relative to the given Yield Curve group. Appended to the last row and the last column of the Data Map is the summation of frequencies of each row and column respectively.

## Probability of a Return (Prior)

Using the Data Map, we can calculate the probability of a given Stock Market Return or group of returns, by summing the frequency range of the returns (sum of rows in the Data Map) and dividing by the total number of observations. For the probability of a bear market ("P[BM]"), we use the standard bear market definition of a decline of 20% or greater. This calculation is called the Prior in the Bayesian Inference method and represents the unconditional probability of our dependent variable.

## Probability of the Yield Curve (Marginal Likelihood)

The probability of each rounded Yield Curve's ("P[YC]") group (in 0.05 increments) is calculated by dividing the Yield Curve group's total frequency (sum of the subject column) and dividing by the total number of observations. This probability is referred to as the Marginal Likelihood in Bayesian terms and is the unconditional probability of our independent variable.

## The Probability of Yield Curve Given Bear Market (Likelihood)

The probability of a Yield Curve group given the observed Bear Market stock market return ("P[YC|BM]") is calculated by dividing the sum of the Yield Curve group's Bear Market Return frequency by the sum of the Bear Market Return rows summations (sum of sums). Bayesian statistics refers to this as the Likelihood.

## The Probability of Bear Market Given Yield Curve (Posterior)

The probability of a Bear Market given the Yield Curve group ("P[BM|YC]") is calculated by dividing the product of the probability of a bear market ("P[BM]") and the probability of the Yield Curve given Bear Market ("P[YC|BM") by the probability of Yield Curve ("P[YC]"). This final calculation is the Posterior in Bayesian Inference ("P[BM|YC]=P[YC|BM]*P[BM]/P[YC]"). From this point, we are able to assess probabilities of Bear Markets based on their past experiences against the Yield Curve's behavior.

## Polynomial Regression

We derive a set of probabilities by extending the Bayesian Inference calculation to the range of Yield Curve groups. In an effort to smooth out this data, we then take a 3rd order polynomial regression of the set of probabilities as the final equation to estimate the Bear Market Probability. The model thus gives us a fair estimate of the probability of a Bear Market provided by how the Yield Curve behaves.