You have been assigned a project, "**Applied Time Series Models for Equity Risk - Simple vs. Exponential**" that will enhance your skills and enable you to think like an analyst.

This project is designed to help you master the art of analyzing and predicting the risk of an equity investment. You'll have the opportunity to work with real-world data and explore the use of different time series models to assess risk. By comparing and contrasting the effectiveness of simple models like rolling standard deviations, and exponentially weighted deviations to more complex models like autoregressive integrated moving average (ARIMA) models, and seasonal decomposition models, you'll gain an in-depth understanding of the different risk assessment methods and their suitability for your requirements.

**Prerequisites**

**Module: **Equities | Modeling Systematic Risk

Financial Books: The Basics of Standard Deviation: A Simple Guide

Lectures: Historical Time-Series Data & Equity Shocks | Time-Series Modeling – Equity Price & Returns

Please carefully read the relevant materials, as they will serve as the foundation for you.

Obtain the historical time-series data for a specific equity stock with a minimum lookback period of 3 years.

Calculate the equity returns for the time series and determine the sample statistic mean and standard deviation of the returns.

Compute the rolling risk (standard deviation) over a 3-year lookback period, considering 252 trading days per year.

Repeat the process in point 2(a) by assigning exponentially decreasing weights, with the most recent observation receiving a weight of 0.06 regardless of economic conditions.

Make a list of observations and assumptions, and explain the difference between rolling standard deviations (2(a)) and rolling exponentially weighted deviations (2(b)).

**Team Collaboration and Selection Requirement:**

for this project, candidates have the flexibility to choose their mode of operation:

**Standalone Mode:** If you feel confident and would like to take on the challenge individually, you're welcome to work on the project as a standalone participant. This will give you an opportunity to showcase your individual strengths and decision-making skills.

**Team Collaboration:** Alternatively, if you believe that collaboration will enhance the quality and depth of your analysis, you are encouraged to form a team. Collaborative efforts often bring diverse perspectives, leading to richer insights and more comprehensive results.

**Self-Selection:**Candidates are free to select their own teammates. If you already have someone in mind, align with them, and inform the project coordinator of your team composition.**Team Size:**While there's no strict limit, we recommend teams of 2-3 members for effective collaboration and equitable distribution of work.**Commitment Agreement:**Ensure that all members of the team are equally committed to the project.

Please note: Whether you choose to work individually or in a team, the assessment criteria will remain consistent. The emphasis will be on the depth of analysis, quality of insights, and presentation of findings.

**Understanding How Analysts Use Time-Series Models for Data Analysis and Forecasting**

Time-series models are basically, statistical tools that are used to analyze and forecast trends and patterns in data over time. Analysts use time-series models to understand and forecast trends, identify patterns and relationships, improve decision-making, and model complex systems because these models can provide valuable insights and help analysts make more informed decisions, and improve the accuracy of their predictions

**Obtain the historical time-series data for a specific equity stock with a minimum lookback period of 3 years.**

Analysts typically start by collecting historical data on the financial variable, such as stock prices or market returns. The data is then transformed and analyzed using statistical techniques to identify patterns and trends. The results of this analysis are used to develop a model that can be used to forecast future values of the variable.

To extract the historical time-series data of an equity stock with a lookback period of at least 3 years, you will need to follow these steps:

Determine the equity stock you want to extract data for. You will need to know the ticker symbol or other identifier for the stock. [for example, the ticker symbol of HDFC Bank Ltd. is "HDFCBANK"]

Choose a source for the historical time-series data. There are many websites and financial databases that offer this type of data, such as Yahoo Finance.

Access the historical time-series data for the equity stock. You may need to create an account or pay a fee to access the data, depending on the source you choose. But, Yahoo Finance is a free database source to access the data from.

Download or export the data to a file that you can use for analysis. The data should include the date, opening price, closing price, and any other relevant information for each trading day in the lookback period.

Import the data into a program or tool that you can use to analyze the time-series data. This may include a spreadsheet program like Excel, a programming language like Python, or a specialized financial analysis tool.

**Calculate the equity returns for a given time series and determine the sample statistic mean and standard deviation of the returns**

Analysts calculate equity returns and compute the mean and standard deviation of these returns to understand the historical performance, compare the performance of different investments, and make predictions about future performance. To do this, they first collect data on the historical performance of the equity investment as we did above, then calculate the returns for each period in the time series. They then use some statistical tools to compute the mean and standard deviation of the returns and analyze and interpret the results to make informed decisions about the investment. The mean and standard deviation can provide insights into the overall level of risk and return associated with the investment.

Read __The Basics of Standard Deviation: A Simple Guide__ for more!

To calculate equity returns across a time series and compute the sample statistic mean and standard deviation, you will need to follow these steps:

Gather the data for the equity returns over the time series. This may include daily, weekly, or monthly returns.

Calculate the equity returns for each time period. You can do this either by calculating absolute returns (by taking the difference between the closing price and opening price), discrete proportional returns (by taking the difference between the closing price and opening price and dividing it by the opening price), or continuous proportional returns (by taking the LN of the resultant value of closing price divided by the opening price).

Compute the mean of the equity returns. You can use the following formula to do this:

Mean Return = SUM(Returns) / COUNT(Returns)

Where returns are the array of daily returns.

Compute the standard deviation of the equity returns. You can use the following formula to do this:

Standard Deviation = SQRT(SUM((Returns - Average(Returns))^2) / (COUNT(Returns) - 1))

Where returns are the array of daily returns, and the count of returns is the number of days (252 in our project).

You can now use the mean and standard deviation to analyze the equity returns and make informed decisions about your investments.

**Compute the rolling risk (rolling standard deviation) over a 3-year lookback period, considering 252 trading days per year.**

Rolling risk (also known as rolling standard deviation) is calculated as the standard deviation of returns over a certain time period, can provide insights into the level of risk associated with an investment, identify trends or patterns in risk, compare the risk of different investments, and make informed decisions about the investment.

To compute rolling risk (rolling standard deviation) over a lookback period of 3 years, you will need to follow these steps:

Gather the data for the past 3 years of trading. You will need to have the daily returns for each of the 252 trading days in each year.

Compute the standard deviation of the daily returns for each year. You can use the following formula or refer:

__The Basics of Standard Deviation: A Simple Guide__

Standard Deviation = SQRT(SUM((Returns - Average(Returns))^2) / (COUNT(Returns) - 1))

Where returns are the array of daily returns, and the count of returns is the number of days (252 in our project).

Repeat the process every trading day to get the rolling risk (standard deviation) over the lookback period for each day.

**Repeat the process in point 2(a) by assigning exponentially decreasing weights, with the most recent observation receiving a weight of 0.06 regardless of economic conditions.**

Another method for calculating rolling risk (rolling standard deviation) is to use exponentially decreasing weights. Exponentially decreasing weights refer to a type of weighting scheme in which the weights assigned to each data point in a time series decrease exponentially as the data points become older. This means that more recent data points are given higher weights, while older data points are given lower weights.

To compute rolling risk (standard deviation) over a lookback period using exponentially decreasing weights, you will need to follow these steps:

Gather the data for the past 3 years of trading. You will need to have the daily returns for each of the 252 trading days in each year.

Compute the standard deviation of the daily returns using exponentially decreasing weights. To do this, you can use the following formula:

EWA Standard Deviation = SQRT(SUMPRODUCT((Returns - Average(Returns))^2, Weights) / SUM(Weights))

Where returns are the array of daily returns, and weights are the array of weights for each day.

In this case, you can use a weight of 0.06 for the most recent observation, and assign exponentially decreasing weights to the other observations as explained in the __Exponentially Weighted Moving Average (EWMA) Approach__. For example, you could assign a weight of 0.06 * 0.94^1 to the second most recent observation, a weight of 0.06 * 0.94^2 to the third most recent observation, and so on.

Repeat the process every trading day to get the rolling risk (standard deviation) over the lookback period for each day.

**Make a list of observations and provide an explanation of the difference between rolling standard deviations (2(a)) and rolling exponentially weighted deviations (2(b)).**

**Observation-1: **Rolling standard deviations (2(a)) are calculated by taking the standard deviation of a set of data over a given lookback period.

for example, if you have a lookback period of 3 years, you would calculate the standard deviation of the data for each year and then average the standard deviations to get the rolling standard deviation.

**Observation-2: **Rolling exponentially weighted deviations (2(b)) are calculated in a similar way, but with the added step of assigning exponentially decreasing weights to the data.

for example, you might assign a weight of 0.06 to the most recent observation and a weight of 0.06 * 0.94^1 to the second most recent observation, and so on.

**Difference-1: **The main difference between rolling standard deviations and rolling exponentially weighted deviations is the way that the data is weighted.

In rolling standard deviations, all data points are given equal weight.

In rolling exponentially weighted deviations, more recent observations are given higher weights and older observations are given lower weights.

**Difference-2: **This difference can affect the resulting standard deviation calculation.

for example, if there is a large change in the data in the most recent observation, it will have a greater impact on the rolling exponentially weighted deviation than it would on the rolling standard deviation. This can make rolling exponentially weighted deviations more reactive to changes in the data.

**Difference-3: **Another difference is that rolling exponentially weighted deviations can be more sensitive to the choice of the decay factor (e.g. 0.94 in the example above). A smaller decay factor will result in higher weights for more recent observations and a larger decay factor will result in lower weights for more recent observations. This can affect the resulting standard deviation calculation and may require more careful consideration when selecting the decay factor.

**Note: **Time series models are based on the assumption that historical data can provide insight into future trends.

**End-Of-Line Discussion**

In addition to the commonly used time series models, there are also more advanced models available such as simple linear models, autoregressive integrated moving average (ARIMA) models, and seasonal decomposition models for analyzing and forecasting time-dependent variables like stock price, volatility, and other risk attributes. We will delve into these advanced models in our upcoming sessions.

This chart indicates the standard deviation and EWA standard deviation of ICICI Bank which we computed rolling risk (STD. DEV.) over a 3 year lock back period assuming 252 days a year as a window provided, considered that the weight is assigned equally to the data.

EWA standard deviation calculated the same way but in here we assigned the exponentially decreasing weight of 0.06 to most recent observation and 0.06*0.94^1 to the second most recent observation and so on, where 0.94 is nothing but a decay factor.

The chart represents the 3 Year Standard deviation and Exponentially weighted standard deviation of UltraTech Cements, from the observation and the chart its is evident that Exponentially weighted Standard deviation is so sensitive to the price movements by giving higher weightage to the recent observation whereas on the other hand Standard deviation giving equal weightage to all the observation. Hereby from the graph we see plateauing in standard deviation whereas in EWMA has no plateauing. For verdict the EMWA has its own advantage compared to the Standard deviation approach, it depends on the investment horizon and type of investment strategy that has been made by the investor.

Observations: We took 3 year historical data form December'2019 to November'2022 of four Automobile Company such as Tata Motors, Maruti Suzuki, Mahindra & Mahindra and Bajaj Auto.1) Equity Return are calculated by the discrete proportion method because there is a gap up in prices of a set of data over a given lock back period and also calculated statistic mean and standard deviation using 'stdv.s' given that the data is sample not the entire population.2) We computed rolling risk (standard deviation) over a 3 year lock back period assuming 252 trading days a year as a window provided for calculating rolling standard deviation considered that the weight is equal assigned to the data.3) EWA standard deviation c…