# Liquidity hoarding and systemic failure in the ecology of banks

As you might have guessed from my recent posts, I am cautious in trying to use mathematics to build insilications for predicting, profiting from, or controlling financial markets. However, I realize the wealth of data available on financial networks and interactions (compared to similar resources in ecology, for example) and the myriad of interesting questions about both economics and humans (and their institutions) more generally that understanding finance can answer. As such, I am more than happy to look at heuristics and other toy models in order to learn about financial systems. I am particularly interested in understanding the interplay between individual versus systemic risk because of analogies to social dilemmas in evolutionary game theory (and the related discussions of individual vs. inclusive vs. group fitness) and recently developed connections with modeling in ecology.

Three-month Libor-overnight Interest Swap based on data from Bloomberg and figure 1 of Domanski & Turner (2011). The vertical line marks 15 September 2008 — the day Lehman Brothers filed for bankruptcy.

A particular interesting phenomenon to understand is the sudden liquidity freeze during the recent financial crisis — interbank lending beyond very short maturities virtually disappeared, three-month Libor (a key benchmarks for interest rates on interbank loans) skyrocketed, and the world banking system ground to a halt. The proximate cause for this phase transition was the bankruptcy of Lehman Brothers — the fourth largest investment bank in the US — at 1:45 am on 15 September 2008, but the real culprit lay in build up of unchecked systemic risk (Ivashina & Scharfstein, 2010; Domanski & Turner, 2011; Gorton & Metrick, 2012). Since I am no economist, banker, or trader, the connections and simple mathematical models that Robert May has been advocating (e.g. May, Levin, & Sugihara (2008)) serve as my window into this foreign land. The idea of a good heuristic model is to cut away all non-essential features and try to capture the essence of the complicated phenomena needed for our insight. In this case, we need to keep around an idealized version of banks, their loan network, some external assets with which to trigger an initial failure, and a way to represent confidence. The question then becomes: under what conditions is the initial failure contained to one or a few banks, and when does it paralyze or — without intervention — destroy the whole financial system?

Arinaminpathy, Kapadia, & May (2012) address this question in a minimal heuristic model. For them a bank is an agent represented by its balance sheet, and the network is a random regular graph with a slight twist: instead of all nodes having the same degree, the banks are divided into two classes with large banks having $\lambda$ times more connections (i.e. loans) and asset value than small banks. The figure below represents a section of this simple banking network. Each bank carries some external risk through its external assets (in the yellow box), and systemic risk through its interbank borrowing and lending (in the red box). The loans are made to other banks in the model, serving as directed edges in to form the lending graph. If you wanted a more empirical basis then you could modify this model by incorporating a real lending network, such as the one from Soramäki et al. (2007).

If the capital buffer of a bank (difference between assets and liabilities; represented by the dotted box in above figure) ever becomes negative then the bank is insolvent and defaults. Combined with the loan network, this produces a coupling of banks through counterparty defaults. If a bank defaults, then its remaining assets (after sale) are divided uniformly among that bank’s creditors. This reduces the lenders capital position (since they only get a fraction of their loan back) and can, by itself, cause a cascade of capital defaults (May & Arinaminpathy, 2010).

Interbank loans all have the same monetary value but come in two forms: short and long. In the liquidity hoarding scenarios, a short loan can be cancelled in just one time step, but a long loan needs one time step to be converted into a short loan (which can then be cancelled on the next time step if desired). All uncancelled loans are rolled over to the next time step. The decision to retain or cancel a loan is based on an assessment of the health of banks. In particular, if a bank i holds a proportion $c_i$ of its initial capital and has a proportion $m_i$ of it assets as liquid (i.e. liquid assets — the region between the yellow and red boxes in above figure — and short term loans) then Arinaminpathy et al. (2012) define its health as $h_i = c_im_i$. A bank’s risk management strategy is given by the conditions under which it shortens its long loans and cancels its short loans, withdrawal can produce a shock propagating from lender to borrower — opposite direction of counterparty defaults. Here, the authors again use an arbitrary rule. A long loan from i to j is made short if $h_ih_j < (1 - C)$ and a short loan is withdrawn (or not rolled over) if $h_ih_j < (1 - C)^2$ where $C = AE$ and A is the proportion of initial assets still in the system, and E is the fraction of interbank loans that are not withdrawn. Of course, these decision process are deliberate (and arbitrarily) simple, and their detail are not particularily essential. What matters is that conditions need to be more dire for a short loan to be withdrawn than for a long loan to be shortened, and that from a local perspective shortening or withdrawing a loan increases the lenders health and makes others less likely to shorter or withdraw the borrowings that lender holds.

However, here we can still see a weakness of the heuristic approach to modeling. Now, I have no problem with the simplicity and lack of empirical basis for the risk management strategy; for the purposes of toy models it is fine to assume some functional form that respects a general property we want risk management strategies to have. What bothers me is that all banks follow the same risk management strategy. Arinaminpathy et al. (2012) are studying system robustness, but in both ecology (Elmqvist et al., 2003) and finance (Beale et al., 2011) it is known that heterogeneity tends to increase system-wide robustness. Hence, using more realistic and diverse risk management strategies is not merely an unreasonable and irrelevant complication to the model, but is known to directly affect the central question the model was built to address. In contrast, if the authors were building an abstraction instead of a heuristic then they would design their model in such a way that more realistic tweaks would leave their conclusions unchanged or only further reinforce them.

A second way to transmit shocks is indirectly through the effect on the prices of external assets, this is the asset shock scenario. If a bank fails and is forced to sell off its assets, then this can drive down the price of this asset class, and lead to losses for other banks holding the same asset class. In particular, following previous work (like Gai & Kapadia, 2010 or May & Arinaminpathy, 2010) they suppose that if a bank is selling a fraction $x_i$ of an asset then that assets price diminishes to a fraction $\exp(-(1 - C)x_i)$ of its original value. Note that while system confidence is high $C \approx 1$, asset liquidation has little effect on the overall price, but if the system confidence decreases then the same proportion sold can lead to a more drastic price drop. However, apart from the overall system wide confidence parameter $C$, the prices of different assets are uncorrelated, and it is assumed that assets are sold to investors external to the system being modeled. Each bank holds some fraction of possible assets (depends on parameter settings), selected at random, with big and small banks holding the same number of assets but big banks just more invested in each asset they hold. If you wanted to build a more data-driven version of this model then it would be worthwhile to follow Huang et al. (2013) and use real asset allocation data.

To introduce the initial shock, Arinaminpathy et al. (2012) consider two possibilities: either the failure of a single bank, or asset. They show (unsurprisingly) that the failure of a big index bank is much more likely to result in systemic failure. However, more surprising is that although the impact of large bank collapse scales with the concentration parameter $\lambda$ without hoarding or asset shocks, and linearly (and only barely faster) with one of the effects turned on, with both hoarding and asset shocks allowed the impact of large index failure increases drastically. In particular, although for $1 \leq \lambda \leq 20$ the impact of large bank collapse scales linearly, in the region $20 \leq \lambda \leq 40$ the effect of large banks increases rapidly, and for $\lambda \geq 40$, the failure of a single large bank almost surely results in the collapse of the whole system. In other words, these banks become too big to fail. Similar results are seen for an initial asset shock, but in these cases more than one bank fail initially.

Finally, the authors explore the effects on controlling the capital ratios that a bank must satisfy (parameter $\gamma$ in the balance sheet figure). The consider placing different capital buffer requirements for big and small banks, and conclude that “contagion is better mitigated by well-capitalized big banks than by well-capitalized small ones: arguable the converse of the pattern of capital ratios before the current financial crisis.” Which highlights the take-home message of this (and most other recent) paper: don’t let self-interested banks set policy for managing system risk, like in any social dilemma, what is best for the individual bank can often be worse for the system.

### References

Arinaminpathy, N., Kapadia, S., & May, R.M. (2012). Size and complexity in model financial systems. Proceedings of the National Academy of Sciences of the United States of America, 109 (45), 18338-18343 PMID: 23091020

Beale N., Rand D.G., Battey H., Croxson K., May R.M. & Nowak M.A. (2011). Individual versus systemic risk and the Regulator’s Dilemma, Proceedings of the National Academy of Sciences USA, 108(31): 12647-12652.

Domanski, D., & Turner, P. (2011). The great liquidity freeze: What does it mean for international banking? ADBI Working Paper No. 291.

Elmqvist, T., Folke, C., Nyström, M., Peterson, G., Bengtsson, J., Walker, B., & Norberg, J. (2003). Response diversity, ecosystem change, and resilience. Frontiers in Ecology and the Environment, 1(9): 488-494.

Gai, P., & Kapadia, S. (2010). Contagion in financial networks. Proc. Royal Soc. A., 466: 2401-2423.

Gorton, G., & Metrick, A. (2012). Securitized banking and the run on repo. Journal of Financial Economics, 104(3): 425-451.

Huang, X., Vodenska, I., Havlin, S., & Stanley, H. E. (2013). Cascading Failures in Bi-partite Graphs: Model for Systemic Risk Propagation. Scientific reports, 3.

Ivashina, V., & Scharfstein, D. (2010). Bank lending during the financial crisis of 2008. Journal of Financial Economics, 97(3): 319-338.

May, R.M., Levin, S.A., & Sugihara, G. (2008). Ecology for bankers. Nature, 451(7181): 893-895.

May, R.M., & Arinaminpathy, N. (2010). Systemic risk: The dynamics of model banking systems. J. Royal Soc. Interface, 7:823-838.

Soramäki, K., Bech, M. L., Arnold, J., Glass, R. J., & Beyeler, W. E. (2007). The topology of interbank payment flows. Physica A: Statistical Mechanics and its Applications, 379(1), 317-333.