Leverage Control -- A Subtle Story

I mentioned recently some work (in progress) by Stefan Thurner and colleagues exploring how leverage influences stability (price volatility) in a competitive, speculative market. Thurner spoke about this at a meeting on Tipping Points in Durham, UK. What I find most appealing about this work is that is explores this question with a model that is rich enough to exhibit many of the basic features we see in speculative markets -- competition between hedge funds and other investment firms to attract investors' funds, the use of leverage to amplify potential gains, the monitoring of leverage by banks who lend to the investment firms, occasional abrupt crashes and bankruptcies, etc.

Is it a perfect model? Of course not, there is no such thing; models are tools for thinking. But it is arguably better than anything else we currently have for running "policy experiments" to test what might happen in such a market if regulators take this or that step -- establishing tight limits to allowed leverage, for example. 

Stefan kindly sent me the slides from his talk, a few of which I'd like to mention here. As I said, this is work in progress, so these are preliminary results. They're interesting because they suggest that avoiding dangerous market instability through leverage limits comes with costs, and that our intuition isn't at all a reliable guide -- we need these kinds of models in which we can discover surprising outcomes (before we discover them in reality).

I won't give a detailed description of the model; it can be found in an early draft of the paper available here. Thurner and colleagues have been working to improve the model over several years, and it now reproduces a number of realistic market behaviors quite naturally. Thurner summarized these as follows:

 

In other words, the hedge funds act to eliminate mis-pricings (taking volatility out of the market), and profit by doing so. Funds have to be aggressive to survive in the face of stuff competition, but suffer if they get too large. Risks shorten the lifetime of a fund. Overall, the models also reproduces the right statistical fluctuations in the market.

As I discussed in my earlier post in this work, competition between hedge funds leads naturally to increasing leverage and drives the market to have a fat-tailed distribution of returns; it becomes subject (like real market) to large price fluctuations as a matter of course driven by its own internal dynamics (no external impacts required). In this condition, the market is highly prone to catastrophic crashes triggered by nothing by small price fluctuations linked to noise traders (unsophisticated investors buying and selling more or less at random). The figure below shows a typical example, plotting the wealth of various funds versus time, with a dramatic crash that affects all funds at once (different colors for different funds):

Now, a natural question is -- could these kinds of events be avoided with proper regulations? One idea would be to restrict the amount of leverage allowed with the aim of keeping the market returns in a mode Gaussian regime, i.e. eliminating fat tails. People could probably argue for decades about whether this would work or not without coming to an answer; this model makes it possible to do an experiment to find out, which is what Thurner and colleagues have done.

Two figures (below) show some of the results, and require some explanation. The different colors correspond to different possible regulatory regimes, and show how behavior changes with maximum allowed hedge fund leverage : BLUE (no other regulations), PALE GREEN (regulations akin to Basel I and II, in which banks loaning to hedge funds are restricted by capital requirements) and RED (a situation in which banks monitor hedge funds and reduce a hedge fund's allowed leverage below the maximum when the volatility in its assets grows; a kind of adaptive leverage control). 

First, consider a figure showing how how the action of hedge funds, and their use of volatility, actually benefits the market -- making it more efficient (in one sense). 

The figure shows the mean square price volatility versus allowed leverage. Increasing leverage lets the hedge funds pounce on opportunities more aggressively and wipe out mis-pricings more effectively. Die hard free market people should love this as it shows that the effect is strongest in the absence of any regulation. The regulated markets require higher leverage to get the same reduction in volatility.

But this isn't the whole story. Now consider another figure for the probability (per unit time) of a failure of one of the hedge funds:

Here the pure free market solution isn't so good, as this probability rises rapidly with increasing leverage. There is a relatively low value of leverage (around 5 in the model's units) where the market benefits of leverage have already been realized, and more leverage only leads to more failures (because it takes the market into the regime of fat-tailed returns; this can happen even if the mean square volatility remains small).

The regulated markets in this case perform marginally better -- the regulations reduce the number of failures, and the cost for this is marginally increased volatility.

A surprising outcome is that these same regulations, in the regime of very high leverage, actually do worse than no regulations at all -- they lead to higher market volatility AND more failures as well, a truly perverse regime.

All in all, then, this model offers a sobering perspective on how regulators might go about trying to avoid crashes linked to fat tails by limiting leverage. Some limitation clearly seems to be good. But too much can be bad, especially when coupled with other market regulations. You can't test out one idea in isolation, because they interact in surprising ways.

I'll probably have some further comments on this in the near future. It's a work in progress, as is my understanding of it -- and of what it means for the bigger picture.

Tax Codes (Yawn!) for Financial Stability?

No one (I hope) enjoys reading about tax codes, but Simon Johnson makes a very good point: they make be potentially very useful in helping to stablize markets.

His reasoning is simple. Any number of studies show that, other things being equal, the use of more leverage by banks, hedge funds and other investors creates more instability -- it can amplify small market fluctuations into far larger market upheavals. So stability would be improved by limiting leverage (although how much to limit it is a matter of some subtlety). You can limit leverage with laws, or with incentives. Johnson is thinking about incentives, particularly through the tax code (in the US). Currently, if a hedge fund seeks leverage by borrowing money, they pay interest on that loan and that interest can be deducted from their taxes. Interest payments are deductible. In contrast, if the same fund raises money by selling shares of its stock, they pay dividends on those shares. Those dividend payments are NOT deductible in US tax law. Hence, investing firms have every incentive to raise money for leverage by borrowing, rather than by selling stock.

An elimination of this tax difference may be one way to attempt to reign in the use of leverage and keep it within the bounds of safety. For those interested in the gritty details, see Johnson's testimony at a recent meeting on (double Yawn!) Tax Reform and the Tax Treatment of Debt and Equity.

Leverage control for market stability

I listened today to a number of extremely informative talks at a workshop in Durham (UK) on Tipping Points in Financial Systems (description here part way down the page). I'll make some comments on the various talks in coming days. But it might be worth noting a few observations on some further progress on a model of market volatility -- and its inherent link to leverage -- achieved by Stefan Thurner and colleagues.

I wrote about this work several years ago in an OpEd for the New York Times, and also in this thing for Nature, but today learned about some further developments which seem particularly important. The model developed in this work makes the point that saavy participants in speculative markets (call them "hedge funds," but they could banks or just one individual) can use leverage to deliver higher returns and thereby attract more investors. This is obvious and natural. Many details aside, however, the model showed that the competition between funds to attract investors drives a race to higher leverage, increasing market volatility, and the eventual probability of violent market crashes. Leverage is dangerous and comes with systemic costs.

This can be seen (in an abstract way, sorry) from the figure below from the paper. In a long simulation of the market, this shows that the likelihood of finding market returns (absolute value of the logarithm of prices differences over a short time) exceeding a value R. The red is how the market works when leverage is low -- the probability to see really big market movements, R > 0.1 or so, is extremely small. But as hedge funds evolve to use significant leverage, the market moves into a regime described instead by the blue curve -- the probability of tail events and extreme movements becomes orders of magnitude larger.

The implication is clear: leverage causes volatility.

But Thurner suggested today that intermediate levels of leverage actually reduce market volatility, because it makes it easier for the saavy hedge fund investors to pounce on and wipe out market mispricings. This is an interesting point and one worth pondering. I haven't yet digested the latter parts of the updated paper, which now considers several policy moves and how they influence volatility, but the results have the wonderful ambiguity that one learns to expect in confronting complex systems. For example, capping leverage at intermediate levels (factors of around 10) is in some case worse than capping it at higher levels (around 15). Controls on the capital reserves held by the funds (or banks) also have some ambiguous results -- in some cases, making them hold higher reserves can lead to more volatility in the market, not less. Weird.

I'll try to digest this new work and report on it's implications once I understand them more clearly, but they already demonstrate the point that our intuition isn't so good at seeing the link between interventions in markets and the likely consequences. I'm certainly guilty on occasion of thinking that if the financial industry is against any proposed regulation, then it must be a good one. Often that's not a bad rule of thumb. But if we're really going to make progress in making markets work for everyone, we need to think very carefully -- and back up proposals with hard evidence. This work is developing such evidence.

How derivatives make markets unstable: Part I

I posted a while back on some of the dirty secrets of the derivatives industry. I promised then to give a little more discussion at some point of two terrifically important pieces of research -- still not widely known, especially in mainstream finance -- which show how adding more derivatives to a market can make it less stable, not more stable. This goes directly against the received wisdom of economic (equilibrium) theory which claims that markets become more efficient as they become more complete, i.e. as it becomes possible to take essentially any kind of market position by virtue of a dense spectrum of financial instruments.

One of the papers I had in mind was this landmark study from several years ago in which William Brock, Cars Hommes and Florian Wagener considered the question of whether, in the run up to the recent crisis, "... highly leveraged positions using complex financial instruments may have amplified market volatility." The answer to which their analysis leads is -- yes, quite probably. More generally, they illustrate how more derivatives in general should make markets more unstable, increasing volatility.

Their paper is a little technical, but worth a read. I'll outline the gist of their argument, which starts with several straightforward observations and moves to a not-so-obvious conclusion:

Observation 1: They start by noting that people aren't the hyper-rational automatons of Milton Friedman's (or other neo-classical economists') favorite fantasies. Rather, people in the real world form their expectations and craft their behaviour in an adaptive way -- that is, they learn from experience.

Observation 2: They also note that people aren't identical. We not only learn, but our brains are different and we've all had different experiences in the past, so, at any moment, we've probably learned different things and have slightly different expectations (heterogeneous expectations, in economic lingo) about the future.

Observation 3: People are generally risk averse -- if they're willing to bet $100 on a gamble that could pay off, but involves risks, they'll be willing to bet more than $100 in the same gamble if you reduce the risks. In other words, people shy away from gambles more the riskier they are. This is basic empirical psychology.

Starting from these observations, Brock and colleagues then consider an "intertemporal" asset market (economist-speak meaning a market in which time exists) in which a lot of people look to past prices and try to predict future prices, buying and selling as they see fit. This market contains both risky and non-risky things to invest in -- stocks and risk-free bonds (which are guaranteed to increase in value by a factor R>1 over each interval of time). Stocks might rise more, but are less certain and hence riskier. In addition, the people can buy derivatives -- instruments which act like pure bets and give a pay off in certain circumstances.

What this all amounts to is that people in this market can 1) play it safe by buying bonds, 2) gamble more by buying stocks, and also 3) buy derivatives if they want which (in this model) have no effect except to offset some of the risks involved in buying stocks.

What Brock and colleagues then show is that the combination of the derivatives, the risk aversion of investors, and their tendency to learn by "reinforcement" -- to be more likely to follow strategies which have paid off in the past -- leads directly to trouble. I'll describe how in a moment, but one final thing before I do: the strength of reinforcement learning in the model (how quickly people shift to use better performing strategies) is controlled by one parameter β; bigger β means faster switching. In previous work, Brock and Hommes have shown that in an asset market in which people learn by the reinforcement process, there is a natural "tipping point" -- at a certain critical value of β -- where the market goes from being stable to being unstable. Intuitively, when people switch too quickly, taking even scanty short term evidence as proof of a strategy's superiority, fluctuations in the market become much stronger.

OK, so what happens in this market when you currently have, say, 15 possible derivatives covering lots of different possible outcomes, and now add a 16th derivative to cover other outcomes (i.e. we have derivatives on stocks and commodities, and suddenly invent some new ones to cover mortgage bonds)? Brock and colleagues show that the addition of this one new derivative makes the market go unstable more quickly, i.e. at a lower value of β. The mechanism involves a simple interplay of reduced risk and human confidence. This new derivative, by making it possible for investors to lower the risks associated with investments, leads them to invest more money. They take bigger bets. These bigger bets naturally amplify how quickly the bets that turn out to be correct amass profits. So, there are bigger differences in the payoffs to recent winning and losing strategies, which draws more followers to the winners more quickly (even if the fundamental switching rate of people haven't changed).

In brief: by the very act of reducing the risk of some strategies, the derivative invites more vigorous gambling on that strategy, leading to faster flows of people from one strategy to another. The extra derivative makes the market more volatile.

This model doesn't involve many questionable assumptions. It's a very basic model of the most central facts of any market, respecting some realities of human psychology. It suggests that derivatives hold inherent dangers. Yet as far as I can see, the ongoing discussion of regulating derivatives isn't taking this perspective into account. As Satyajit Das notes, the drive toward greater returns that is an essential part of the dynamics in the Brock, Hommes and Wagener model is a very real force in today's derivatives markets:

Investors searching for return drive speculation. Concerned about stagnant real incomes and inadequate retirement savings, individual investors seek out higher yielding investment structures, often based on derivatives. Pension funds and other institutional investors use derivatives to enhance returns to fully fund and meet their contracted liabilities. In an environment of diminishing returns and fierce competition for attractive investments, fund managers use derivative strategies to enhance returns through readily accessible leverage and capacity to create risk “cocktails”.

Facing increased pressure on earnings, corporations have increasingly “financialised”, resorting to speculative derivative trading to meet profit expectations. ... [Such] seculative activity amplifies rather than reduces volatility and systemic risks. Perversely, this may impede capital formation and also increase the cost of capital for companies.

What happens in the real world backs up the lesson of this simple model. Derivatives reduce risks only in a very narrow and restricted sense, while undermining the functioning of markets more generally. Of course, there's lots of money to be made by the people selling derivatives, so don't expect them to admit (or care about) any of this.