Risk Control

This section gives a more detailed breakdown of the functionality that is available in the risk control section of the website.

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 covariance 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 covariance matrix may be observed by increases in the Burg Divergence.

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.

Ledoit Shrinkage

Ledoit Shrinkage is based on the assumption that the sample covariance matrix, i.e the covariance matrix constructed by those points in the overall timeseries that are used, is a bad approximator of the "real" covariance matrix. The principal objection is that the sample covariance matrix is very volatile so that the assumption that it resembles the real covariance matrix is unlikely to be a good one. In the place of the sample covariance matrix it uses a process that "shrinks" the values of co-efficients that is considers outliers, closer to values that it considers central. As a result, the shrunk covariance matrix should exhibit more stability than the sample covariance matrix over time. This may be a good thing.

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.

Passive Hedging

Passive Hedging is the simplest type of risk control. It attempts to detemine the most efficient manner of reducing risk in the portfolio utilising a user-defined set of hedging instruments. There are no other contraints, other the the type and number of hedging instruments available, and no information is given on expected drift or volatility of the hedging instruments. The analysis may use either a multi-variate least squares optimiser or it is possible to optimise over a specific risk metric. Please note however, in the case that the risk metric is not coherent, there is a non-zero probability that the suggested hedge portfolio is not a global minimum.

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.