Time Series for Actuaries

What Is a Time Series?

A time series is a sequence of data points that appear in sequential order over a period of time. This can be compare with cross-sectional data, which captures a point-in-time.

A time series follows the motion of the chosen data points, then the period of time with data points set down the regular gaps. There is no lowest or highest amount of time that must be involve, allowing the data to be assemble in a way that provides the information being sought by the investor or analyst examining the activity.

What is an actuary?

Actuaries consume a unique combination of mathematical, communication, analytical and management skills. They use their skills to create social impact, to make high-level strategic decisions, and have a significant impact on the law, business and people’s lives.

Actuaries are curious, creative and adaptable — a mentality that helps them succeed in the digital age.

Shareholders’ unique combination of technical skills and professional skills ensures that they continue to do their part, protecting themselves from the impact of future uncertainties.

Understanding Time Series :

A time series can be considered for any variable that changes over time. After investing, it is common to use to track the price of a security over time. This can be achieved in the short term, like the price of a security at that time on a business day, or in the long term, like the price of a security at the end of the last day of every month for five years.

The Stationary and Markov Property :

Stationary:

A stationary process has the property that the structure of the mean, variance and autocorrelation does not change over time. Stationarity can be defined in precise mathematical terms, but for our purposes we mean a series with a flat appearance, no trend, constant variance over time, structure of constant autocorrelation over time, and no periodic instability.

Markov:

The Markov processes are an important class of the stochastic processes. The Markov property means that the evolution of the Markov process in the future depends only on the current state and does not depend on the past history. The Markov process does not remember the past if the present state is given. Hence, the Markov process is called the process with memoryless property. The theory of these types of processes allows us to create real random process models, especially in terms of reliability and serviceability.

Partial Autocorrelation Function :

Partial autocorrelation is a summary of the relationship between an observation from a time series with observations at previous time steps, with intermediate observation relationships excluded.

Autocorrelation for observation and observation at an earlier point in time consists of direct and indirect correlation. These indirect correlations are a linear function of the correlation of observations with observations at intermediate time intervals.

These are indirect correlations that the partial autocorrelation function tries to eliminate. Without going into math, this is the intuition for partial autocorrelation.

The example below calculates and builds a partial autocorrelation function for the first 40 delays of the minimum daytime temperatures.

ARIMA :

Integrated Autoregressive Moving Average or ARIMA is a statistical analysis model that uses data from time series to better understand a dataset or to predict future trends. The statistical model is autoregressive if it predicts the future values based on early values.

What is ARIMA used for?

ARIMA is a method of predicting or predicting future results based on historical series. It is based on the statistical concept of sequential correlation where past data points affect future data points.

Autoregressive(AR):

This time series model can be uses for observations from previous time steps as input to a regression equation to predict the value at the next time step.

Integrated (I):

Integrated is the differentiation of raw observations that allows the time series to become stationary (i.e., the data values are replaced by the difference between the data values and the previous values).

Moving average (MA):

A moving average includes the relationship between an observation and the residual error of a moving average model applied to pending observations.

ARIMA Parameters:

Each ARIMA component functions can use as a parameter with a standard designation. For ARIMA models, the default is ARIMA notation with p, d, and q, where integer values replace parameters to indicate the type of ARIMA model to use.

GARCH models for measuring volatility:

The Generalized Process of Autoregressive Conditional Heteroscedasticity (GARCH) is an econometric term coined in 1982 by Robert F. Angle, economist and Nobel laureate in economics in 2003. GARCH describes an approach to assessing volatility in financial markets.

There are several forms of GARCH modeling. Financial professionals often prefer the GARCH process because it provides a more realistic context than other models when trying to predict the prices and rates of financial instruments.

KEY TAKEAWAYS:

The generalized process of autoregressive conditional heteroscedasticity (GARCH) is an approach to assessing financial market volatility.

Financial institutions use this model to assess the volatility of returns on stocks, bonds, and other investment vehicles.

The GARCH process provides a more realistic context than other models when predicting prices and rates of financial instruments.

Understanding the GARCH Process :

The general process for the GARCH model includes three steps. The first is to evaluate the most appropriate autoregressive model. The second is to compute the autocorrelation of the error term. The third step is to check the value.

Two other approaches widely used for estimating and forecasting financial volatility are the classic historical volatility method (VolSD) and the volatility exponentially weighted moving average (VolEWMA) method.

Example of the GARCH Process:

GARCH models describe financial markets in which volatility can change, becoming more volatile during a financial crisis or global events, and less volatile during times of relatively calm and stable economic growth. For example, on a chart of returns, stock returns may appear relatively stable in the years leading up to a financial crisis like 2007.

However, in the period after the onset of the crisis, the yield can fluctuate widely from negative to positive. In addition, increased volatility can be an indicator of future volatility. Then volatility may return to pre-crisis levels or may become more even in the future. A simple regression model does not account for this variation in volatility in financial markets. It is not representative of the Black Swan events that occur more frequently than anticipated.