Introduction

Mean reversion is a fundamental concept in time series analysis that describes the tendency of a series to move back toward its historical mean or average over time. It suggests that extreme deviations from the mean are likely to be followed by corrections that bring the series closer to its mean value. Mean reversion is often observed in financial markets and various economic indicators. Understanding mean reversion is essential for forecasting future values and making informed investment decisions.

Mathematical Description

Mean reversion can be mathematically described using the following formula:

$$Y_t=\mu+\epsilon_t$$ Where:

• $Y_t$ is the value of the series at time $t$
• $\mu$ is the long-term mean or equilibrium level of the series
• $\epsilon_t$ represents the random fluctuation or error term at time $t$

The mean-reverting level $(\mu)$ acts as an attractor, pulling the series back toward its longterm average. This concept is particularly relevant when analyzing time series data that exhibit short-term fluctuations around a stable equilibrium.

Calculation of Mean-Reverting Level

To calculate the mean-reverting level ($\mu$) of a time series, follow these steps:

1) Collect Data: Gather historical data of the time series you want to analyze.

2) Calculate Historical Mean: Compute the mean (average) of the historical data.

3) Determine Decay or Reversion Speed: Analyze the speed at which the series reverts to its mean. This may involve statistical techniques such as exponential smoothing or fitting a decay model.

4) Calculate Mean-Reverting Level $(\mu)$: Based on the historical mean and the chosen decay or reversion speed, calculate the mean-reverting level using relevant formulas. For example, in cases of exponential decay, the mean-reverting level might be calculated as:

$$\mu= \text{Historical Mean} \times \text{(1-Decay Factor)}$$

Example: Consider a stock price that tends to revert to its historical average. If the historical mean of the stock price is 100 USD and the decay factor is 0.05 (implying a $5 \%$ decrease per period), the mean-reverting level $(\mu)$ can be calculated as follows:

$$\mu=100 \times(1-0.05)=95$$

In this example, the stock price tends to revert to a mean-reverting level of 95 USD over time.