*“No one is so brave that he is not disturbed by something unexpected.” Julius Caesar*

The last two weeks of market activity jolted important markets. In earlier posts I highlighted a potential early sign of disturbance may have come from copper. Be that as it may, here I look at a comprehensive quantitative measure of financial system behaviour which is yet to register much shock, though many of its constituents certainly moved dramatically and remain somewhat dislocated. That also, unfortunately, does not guarantee that recent shocks are not followed up by more disturbance. But so far, there is little evidence of systemic behaviour.

It is useful to take an Information Theoretic interpretation of global financial markets. Individual market behaviour is a combination of mutual (system) and conditional (idiosyncratic) entropy (price variance). The higher the entropy, the greater the information needed to describe the system and the greater the uncertainty in the system.

Counter-intuitively, greater entropy in the financial system is associated with ‘risk-on’. This makes sense when one realises the *highest* *certainty* in financial system occurs during financial crises, or ‘risk-off’ episodes when correlations between normally different markets move closer together, reducing the information content of the system, so reducing entropy.

It is possible to assign meaning to the main historical contributors to global mutual entropy with a degree of confidence. These are, in order of importance,

Equity risk (bankruptcy risk),

Dollar demand,

Interest rate risk,

Commodity risk.

There are important contributors to entropy in the orders above these 4 dimensions. Higher orders contributions become increasingly noisy as conditional entropy becomes more important, making assignment of economic description uncertain. That said, a reliable understanding of major market variance may include higher order contributors, as conditional entropy occasionally becomes large enough to affect the entire system – for instance, when an individual market shows prices spikes which disturb investment in other markets. This becomes apparent below.

#### A global financial market index based on entropy

A challenge in determining systematic information at a global market level is that there is no recognised index to provide ‘cohesion’. As suggested above, it is possible to construct an index which charts the evolution of global financial entropy and so acts as a global market financial index. This can be done using singular value decomposition as detailed in Caraiani (2014)1, based on ideas suggested by Shannon (1948)2.

**Singular value decomposition identifies entropy**

Any matrix 𝐴(𝑚𝑥𝑛) can be decomposed using the singular value decomposition as:

with 𝑈 an 𝑚𝑥𝑘 matrix and 𝑉 an 𝑛𝑥𝑘 matrix. 𝑆 is a diagonal matrix defined by:

where 𝑘=𝑚𝑖𝑛(𝑚,𝑛). The values of the matrix 𝑆 are nonnegative and ordered from the largest to the smallest elements.

We construct a index of entropy in the global financial system, using the singular values λk.

The result is a time varying measure of entropy for daily data based on a moving window of 12 months.

#### Calibrating the index

The choice of number of λ to include needs calibration to gauge how many orders of entropy to include.

A potential metric to benchmark the right number inputs are the results of non-conventional policy actions by central banks since the GFC. We notice that the highest ~15-30% sources of entropy produces relatively stable series whose peaks and troughs broadly align with changes in reserve balances of global central bank denominated in current dollars in the last 20 years. In other words, the entropy contained in strongest entropy signals in the financial universe seems aligned in some ways to stress recognised by central banks. Well, duh! That is not a surprise: major financial disturbances (reflected in entropy) in the last 20 years have been met repeatedly with:

demand for dollars and

an increase in central bank balance sheet size, funded by reserves.

Although there is a question over how reserves will feature in future as a policy tool, for now, this seems an appropriate metric with which to calibrate entropy over the past decade. If necessary, future recalibration may have to rely on some other metric.

There is no particular magic about this process of calibration. It simply offers a benchmark against which to chose how many λ inputs to allow. The results are similar for any sample of inputs from 4 to 15.

To match the 12 month window used to calculate the entropy, we use a 12 month log change in reserves.

Here are the results.

#### Little evidence of predictive power.

Caraiani argued that entropy contained a predictive capacity with regard to stock market dynamics. In contrast, the lead/lag correlations of global market entropy suggest modestly reactive behaviour from markets to central bank balance sheet expansion.

This is of little consequence for the usefulness of the index which can be used as a gauge of behaviour of its constituent markets. It may suggest that shocks are not immediately transmitted through the entire global financial system. However, that is not our immediate focus.

#### Correspondence to ‘Fear Index’

The Entropy Index as described shows a clear correspondence to the VIX index, commonly known as the ‘Fear Index’. The Entropy Index, though, shows a smoother path and so helpful to identify constituent markets as ‘rich’ or ‘cheap’ relative to the cost of insuring a stock portfolio via the VIX.

It is also the case that spikes in the VIX are generally associated with rising entropy over a longer horizon. The recent sharp rise in VIX may precede a rise in ‘risk-off’ behaviour. As we show below, so far there is little sign of any serious system disruption.

#### Relevance to recent market disturbance

The unwind of the JPY ‘carry trade’ in late July affected some high profile markets, such as NASDAQ and NIKKEI, as well as VIX. Not surprisingly, comment focussed on the behaviour of these supposed ‘bellwether’ indicators rather than on the financial system as a whole.

As the chart below shows, there has been little overall disturbance to the Entropy Index, yet.

Instead, the shock caused different reactions from different parts of the financial universe. This means there are relative laggards as well as notable movers. For fun, we’ve listed a random sample of recent valuations relative to the entropy index showing the high dispersion – the sample differs every time we run the code, so it is given as it appears. And yes, copper appears, close to fair value.

The predictive power of singular value decomposition entropy for stock market dynamics, Petre Caraiani, 2014

A mathematical theory of communication, Claude Shannon, Bell System Technical Journal 27, 1948