Directional entropy and tail uncertainty, with applications to financial hazard

Abstract

“Mine is a long and sad tale”, said the Mouse, turning to Alice and sighing. “It is a long tail certainly,” said Alice, looking down with wonder at the Mouse's tail; “but why do you call it sad?” And she kept on puzzling about it while the mouse was speaking …

Financial risk management metrics such as value at risk (VaR) can be illuminated by means of a regime-specific concept of directional entropy. This enables a change of measure via a rescaling function to an equivalent logistic distribution, one that has the same total and directional entropies at the chosen critical point. VaR rescaling adjusts the critical probability to capture the long tail entropy. The scaling function can be used as a comparative metric for tail length, or equivalent conditional value at risk, even where moments do not exist. Directional entropy can also be used to identify regions of maximal exposure to new information, which can actually increase entropy rather than collapse it.

Keywords:

• Alternative investments
• Applied finance
• Applied mathematical finance
• Applied investment analysis
• Asset liability modelling
• Asset allocation
• Asset management
• Asset pricing

Acknowledgements

Thanks go to seminar participants at the Universities of South Australia and Ulm, also to QF referees for comments that improved the paper's focus.

Notes

† On the other hand, Bowden (2006) has pointed out an inconsistency between VaR and CVaR, in that tolerances for the one cannot be set independently of the other.

‡As another potential application, it could be noted that risk management tools such as VaR or CVaR (collectively, generalised value at risk GVaR) can be viewed as analogous to put options written by the manager (Artzner et al. 1999, Bowden 2006). Similar remarks to the above can therefore apply to call options and warrants that benefit in this case from exposure to the upper tail. Venture capital investments are very often structured in just this form, essentially as real options. The task in such contexts becomes one of identifying investments that benefit from upper directional entropy, including choosing strike prices that will result in maximal exposure to changes in entropy as information unfolds (see also section 5).

†The convention in statistics is to refer to a distribution that contains both discrete and continuous values (jumps in F(x)) as ‘mixed’; while linear combinations of distribution functions are said to generate ‘mixtures’ of distributions.

† The two components are not the same as the two half integrals implicit in the expression (4) for differential entropy. The relationship is given by

with a complementary definition for the other half integral . Likewise, is not the same as the differential entropy of the conditional distribution for x ≤ X.

† Intuitively, for X values greater than the median, the local logistic has to be centred well to the right of the median so that F_g (X) at X coincides with F(X). This means that the CVaR for F will contain values of X to the right of the median, to an increasing extent as X becomes larger.