## Dust flux, Vostok ice core

Two dimensional phase space reconstruction of dust flux from the Vostok core over the period 186-4 ka using the time derivative method. Dust flux on the x-axis, rate of change is on the y-axis. From Gipp (2001).

## Friday, February 24, 2012

### Another view on default cascades--Battiston et al. (2011)

This paper (pdf) was recently published in Switzerland, and provides an interesting look at our recent topic--default cascades. Although these papers are mathematically dense, they are worth working through sometimes as they may give some foreshadowing of future economic policy.

Block-slider model of earthquakes

Battiston et al. (2011) have presented a model of the financial system which might look like one of Turcotte's slider-block models of earthquakes, which are comprised of numerous blocks of (possibly varying) masses, connected by springs, having to slide across a surface with a limited (and possibly variable) friction. Motion in one block can change the stress field across the model, possibly triggering slip in one or more other blocks.

The original slider-block model consisted of two blocks connected by a spring, both of which sat on a somewhat rough surface (so there would be friction between it and the blocks). If block A moves some small distance, then it will add to the forces on block B. That force may be enough to overcome the friction which kept block B stable. If both blocks move together, we have a larger earthquake. The simple two-block slider model exhibits chaotic behaviour (Turcotte, 1997). I remember attending a conference a few years before the above volume was published when Turcotte presented a more advanced model that looked something like the one below.

We are looking at a plan view of several interconnected blocks. The frictional forces vary for each block, and each block has its own driver. Once again, the slippage of a single block may trigger slippages in one or more blocs--the more blocks that slip, the larger the earthquake. We might expect such models to satisfy the Gutenberg-Richter law which is an observed distribution of earthquake sizes through time that is consistent with a system at self-organized criticality (SOC). But I'm not sure because I've never seen the results although comments on similar models used to study avalanches were consistent with SOC (there are those avalanches again).

According to Battiston et al. (2011), prior to the financial crisis of 2008, existing models suggested that major financial entities had diversified their debts and obligations sufficiently that the likelihood of systemic failure was negligible. The observed financial crisis suggests that this conclusion was unwarranted, to say the least. The authors attempt to study the effects of diversification on systemic risks using a model conceptually similar to the block-slider model above.*

In the financial model, the blocks represent financial institutions. There are a large number of possible interactions between one institution and its neighbours. Furthermore, there is a richness to the interactions that is missing in the earthquake slider-block model--the debts and credits between institutions may each be long- or short-dated, so that there may be a mismatch in maturities between the credits and obligations of any one institution.

In the above figure, which shows only a portion of the potential interactions among entities A, h1, h2, etc., the arrows point in the direction in which credit has been extended. Credit may be long- or short-term. For instance, entity A has extended long-term credit to entity j1, and short-term credit to entity m1; and in turn has borrowed long-term from entity h1, and borrowed short-term from entity n1.

The authors carry out the following experiment. Assume an initial allocation of assets and liabilities across different participants, and derive (logically rather than empirically) a law of "motion" related to financial robustness of each agent affected by one or more of the initial defaults, as measured by their equity ratio. Models are run and the size of the default cascade is compared to the initial distribution of robustness and risk diversification.

The interrelationships between all the balance sheets of the various financial institutions links the dynamics of the individual equity ratios in ways that are not easily predictable.

The authors identify two "externalities" to the triggers for default cascades: 1) variability of financial robustness of all of the interconnected financial entities; and 2) the average financial robustness of the interconnected entities.

If all parties have similar financial robustness (variability is low), then increasing connectivity makes the system more robust. Stability is even likely through diversification if the individual parties are not very robust. It was only when the initial robustness was highly variable across agents (i.e., some agents are weak and others strong) that increasing interconnectedness tended to stimulate systemic defaults.

The second "externality" is a consequence of incomplete information--and deals with the likelihood that creditors will force a foreclosure on an otherwise solvent entity due to the fear that some of its counterparties might fail. Losses may therefore be amplified along the chain if runs begin on entities which may be technically solvent, but which may then be forced to sell long-dated assets at fire-sale prices to raise cash. Model runs suggest that if the average robustness of agents is high, then increased connectivity is beneficial. For low levels of average robustness, then increased connectivity has no effect. For intermediate values of average financial robustness, increased connectivity tended to stimulate systemic defaults.

The lesson here is diversification is not always a good idea. If you diversify across financial entities with wide risk profiles (i.e., some are weak and some are strong) you actually increase the likelihood of a financial calamity.

We don't have to confine ourselves to financial institutions. If we consider our agents to be sovereign, we expect the same problem. Creating a financial superpower out of a group of Germanys would be perfect--even a group of Greeces might be okay. But creating one out of Germanys and Greeces tends to encourage a financial catastrophe. Who could have predicted that?

The authors suggest that the "fix" for this situation is to concentrate risk rather than diversify it. I wonder--in whose hands will the risk be concentrated? Perhaps if you hold gold, the risk won't find its way into yours.

References

Battiston, S., Delli Gatti, D., Greenwald, B., and Stiglitz, J. E., 2011. Default cascades: When does risk diversification increase stability? ETH Risk Center Working Paper Series.

Turcotte, D. L., 1997. Fractals and chaos in geology and geophysics, 2nd edition. Cambridge University Press.

* one key difference between the default cascade and an earthquake--in an earthquake, the tsunami (if there is one) happens afterwards. The ocean of liquidity in which we find ourselves has preceded the major financial earthquake.

#### 1 comment:

1. Turcotte's interconnected block model is similar to others I have seen in systems having viscous elements (pneumatic actuation, aeroelastic structures, etc) where springs would be replaced by resistor/damper connections. Life is a large set of interconnected blocks.