Frequently Asked Questions - Time Series Analysis

Time Series Analysis

      This section gives a more detailed breakdown of the functionality that is available in the time series analysis section of the website. Please note that no information is provided on technical analysis metrics. No comment, adverse or otherwise, should be implied from this approach. It is rather that from a risk analysis perspective, it is prefereable that the metrics have at least a modicum of statistical significance associated with them. In the non-linear, non-stationary world of finacial time series, even that may be asking for too much.

Burg Divergence

The Burg divergence is used to study the evolution of the covariance matrix of the portfolio. When one constructs a portfolio, one of the key measures that is used is the expected volatility of the portfolio. If one looks through the portfolio to its individual components, this is effectively represented as the co-variance matrix of the portfolio. There is no attempt here to deny the manifest benefit of diversification. However what starts off as a diversified portfolio may lose that diversification over time. Whilst this may not be apparent in either the performance or the volatility at an aggregate portfolio level, changes in the underlying co-variance matrix may be observed by increases in the Burg Divergence.

Correlation

From a time series perspective the correlation , or Pearson Correlation Coefficient is a normalised covariance between two time series. The normalising factor is normally the product of the two volatilities. It may be thought of as a measure of the degree with which two time series move together. A correlation of 1 implies that they are in lockstep (although their amplitudes may be different), -1 implies that they are moving absoluteley in opposite directions as a correlation of zero implies not discernible relation between the two time series. It should be noted that for small samples, the sample error in correlation estimation may be very large. In addition if one, or both, of the time series are not stationary, the estimated correlation may be a good reflection of the actual correlation between the two time series.

Drift

The drift is a measure of the average value of the time series. It has particular use when looking at more volatile components of a time series, such as Diff and Return, where there is little obvious structure on a daily basis, but there may be some more observable longer term average behaviour

Kullback-Leibler Divergence

The Kullback-Leibler divergence is a measure of the distance between two multivariate normal probability distributions. In the specific case here, it is metric that looks at the change from the ex-ante portfolio to the realised portfolio. It takes into account both drift and co-vartiance and as such may be thoguth of as a weighted combination of the Mahalanobis distance and the Burg divergence. It is widely used in information theory and signal processing to determine how badly a signal has been degraded. It's use in a portfolio is to determine how rae from its initial specification a portfolio has moved.

Kurtosis

The kurtosis of a distribution is a measure of the degree to which a time series experiences large moves as a proportion of the number of small moves. The normal distribution gives a standard reference point and those which have a greater number of large moves to small moves are knows as leptokurtic or fat tailed. Those that have a smaller number are known as platykurtic or thin tailed. The kurtosis is important from a risk perspective in that a large kurtosis implies a large number of large moves that are important for confidence interval metrics such as Value at Risk and Economic Capital.

Mahalanobis Distance

      Mahalanobis distance is the matrix equivalent of the Sharpe ratio, and curiously enough, it has been around for much longer. Effectively it looks at the difference between the ex-post and ex-ante drift of the components of the portfolio, but modified by the inverse of the ex-ante covariance matrix. As such it is a measure of the current portfolio performance compared to the its historical performance. It has a particularly useful property that the probability distribution of the distance is Chi-Squared and so a statistical likelihood may be estimated for all realisations of the parameter.

OutSample Type

       The matrix analysis methods determine "distances" between two sets of time series. The first time series is determined from the size of the InSample (IS) window. The second time series however need to be considered quite carefully. The functionality provided here gives three choices for the size of the OutSample (OS) window and for its positioning in relation to the InSample window. 1. External: In this case the OS window is directly after the IS window. This is most suitable for the Mahalanobis distance because there is no need to determine the covariance matrix of the OS window for this metric. 2. Int + Ext, t-0 = 0: In this case, the IS window and the OS window have the same start point, however the OS window is longer than the IS window by an amount equal to the OutSample Period. 3. Int + Ext, t-0 = OSP: In this case the IS and OS windows are of the same length with the OS window starting and ending the OutSample Period after the IS window.

Sharpe

Sharpe is short for Sharpe Ratio. This is a measure, first introduced by William Sharpe that may be used to estimate the "quality" or a set of returns. It is more normally defined as the ratio of the excess returns to the realised volatility, however, here it is the ratio of the realised return to the realised volatility. In order to ensure consistency the Sharpe that is produced is always an annualised number.

Skew

The Skew of a time series is the third moment of the distribution. This is a measure of the asymmetry of a distribution insofar as the tail on one side of the distribution is longer than the tail on the other side. Obviously the two sides are balanced on either side of the mean, but from a risk prespective the skew indicates whether there is an expectation of larger moves down or up. Generally financial time series have a negative skew. Please note that a Normal distribution (0,1) has a skew of 3 and as such any skews less than this number would be considered negative. In order to regularise this situation, most calculation engines, including this one, subtract 3 from the calculated number so that skews whose values are less than zero, are in fact representative of distributions with negative skews.

Sum

The Sum is what is says. It is the arithmetic sum of the underlying time series over the specified window period.

Volatility

The volatility is the sample volatility, using standard deviation as an estimate, over the specified time period.

Von Neumann Entropy

Von Neumann introduced this quantity, also known as quantum relative entropy in an attempt to extend the classical notion of entropy to quantised, or discrete states. As a system becomes more disorderly, its entropy is said to increase. If you consider that the state of your portfolio at inception to be the most orderly, which can be thought of as saying that the statistical properties of the portfolio are in line with your expectations, then as the system moves away from this original state, from a statistical properties perspective, its Von Neumann entropy will increase. As such this functionality may be used as a diagnostic, along with the Kullback-Leibler divergence to study the stability of the portfolio over time.

Window Functions

      A financial time series is a sample of the totality of the complete time series of the underlying asset. Taking a single year's worth of data for a particular asset, it is effectively the same as having the complete time series and putting all the values to zero for all time before the start of the year, and to zero for all time periods after the end of the particular year. Thus, whilst it may be thought of as a one year unwiegthed time series, in reality it is a time series by which a window function has been multiplied. The function has value zero before and after the year and a value of one during the year. This is know as a Dirichlet window. Window functions has been an integral part of time series analysis since the 19th century, they just never seem to have made it into financial econometrics. I guess that physics envy never extended to fourier envy...However the value of window functions may be seen in comparing the evolution of different statistics over time. Using a Dirichlet window can mean that the volatiltiy of what is meant to be an aggregating statistic is of the same order of magnitude as that of the underlying time series. Howver the drawback is that apart from the Dirichlet window, that apply weights to the points in a time series. Unless you are observing normalised statistics such as correlation, it is difficult to compare values across different window functions.