On The (in)Stability of Stablecoins

A “stablecoin” is an attempt to create a synthetic asset that is “more stable” than its underlying constituents. In this article we prove that the only workable stablecoins are those that are composed of a simple (weighted) average of assets, more commonly called a “basket” or simply, “diversification of your portfolio”.

While stablecoins are on the forefront of the mind of crypto enthusiasts, and bolstered by the success of Tether, this is not a new idea. There are currently 66 countries that peg their currency to the US dollar and 25 that peg theirs to the Euro. The motivation is understandable, smaller countries having national currencies would naturally have higher volatility than larger markets, hurting importers and exporters in their country. By tying their currencies to a larger market with a responsible fiscal policy, they achieve stability for their own importers and exporters, and a more favorable balance of trade.

(Dollar) currency pegs operate by buying and selling US Treasury Bonds (also called T-bills). However if a large market movement causes the pegged currencies government to run out of bonds, they will be unable to maintain the peg. When this happens the only recourse the government has to pay debts is to print more money, leading to hyperinflation. In other words, in such circumstances, the pegged currency is not fully collateralized. While governments typically operate currency pegs manually, crypto stablecoin enthusiasts often propose algorithmic solutions. The fact that a peg is algorithmic doesn’t really matter, since any human intervention bond purchasing/selling strategy can be turned into an automatic algorithm. The point of this article is to show that no algorithm exists that can maintain a peg. Argentina, Mexico, Greece and Thailand are countries which learned this lesson the hard way.

The crucial observation that we will use in this analysis is that financial markets are fat-tailed. That is, the probability of large price movements is much larger than people generally assume. While the probability is finite and can be calculated, the size of the expected movements is infinite, and therefore the assumption that the price will remain within any given trading range is false.

We can model financial markets in terms of probability distributions between two different time intervals. That is, if the price is p right now at time t, the probability distribution gives the price at the next time interval t+1, where 1 can be 1 hour, 1 day, or whatever time interval you desire. This is known as the 1-point correlation function, and some fat-tailed distributions for it are the Cauchy (a.k.a. Lorentz distribution among physicists because it describes particle decays), Lévy, and Frechet distributions.

The alpha-stable Lévy distribution for several values of its variance parameter α. When α=2 this distribution is a Gaussian (normal) distribution; when α=1 it is the Cauchy distribution. [calculation and graph by the author]

In mathematical terms, the second and higher moments of the price probability distributions are infinite. Vilfred Pareto famously used the above alpha-stable Lévy distribution with 1 < α < 2 as a better model of stock and commodity prices than the Gaussian distribution. Use of the Gaussian assumption in the Black-Sholes model of options pricing is well known to under-price options that are far out of the money for instance. Similarly, metrics based on Gaussian statistics such as Bollinger Bands are spectacularly useless during large market movements, and large market movements happen far more often than people intuitively assume.

Now let us turn to a hypothetical “peg” algorithm and see how it fails due to fat-tailed distributions. We assume that the central bank (or stablecoin) will buy or sell the asset to which it is pegged whenever the price moves outside a defined range. In mathematical terms, we can expand a (non-fat-tailed) distribution using the method of moments. (“Moments” are expectation values derived from the distribution, the first three of which are more commonly known as the mean, variance, and skewness)

Let us assume that we want the pegged currency or stablecoin to remain in a range a < u< b relative to the underlying u, for instance let’s assume we want our pegged currency to remain within 1%=(u-a)/u=-(u-b)/u of the underlying. Let’s assume the market drops below this range and the central bank must buy the underlying to compensate. How much of the underlying needs to be purchased depends on the price. The expected price is given by the variance of the distribution, which is infinite for fat-tailed distributions. In the worst case scenario, the pegged currency drops to zero, which requires the central bank or stablecoin to buy back its entire reserves, which requires that the central bank or stable coin have a reserve equal to its entire market cap in the underlying.

Therefore the notion that an algorithmic stablecoin will work with an amount of collateral in the underlying less than the full market cap of the pegged currency is categorically false. If you’re holding the entire market cap in reserve however, this is “fully collateralized,” and no buying/selling is required, rather one only needs to issue new pegged units in concert with acquisition of the underlying.

The notion of a fully collateralized stablecoin has been explored and is often called “issuing a bond on a blockchain” or sometimes “tokenization”. If assets are on deposit in an appropriately regulated and audited manner, it is straightforward to issue new units of the pegged asset in correspondance with bond issuance or deposits. The key phrase here is “appropriately regulated and audited”. Any form of fractional reserve in these deposits creates systemic risk that the peg will fail. It is the audit and trust in that audit and regulatory regime that creates certainty.

Financial firms are famous for lending out collateral on their balance sheet, sometimes leading to events like rehypothecation and commingling. Rehypothecation is the practice of multiple parties claiming the same asset on their balance sheet, for instance the loaner and borrower both counting the assets, which results in a double counting. This creates systemic risk when market conditions force a loan to be recalled. Commingling is the practice of substituting one asset for another in audits. This creates systemic risk when the pledged alternative asset cannot be bought or sold at the expected price.

Financial firms don’t like large amounts of capital sitting on their balance sheet, often called “trapped capital”, since it is perceived that this capital could be more productive if loaned out. However this is the promise of crypto-currency — everything is fully and systematically capitalized all the way through with cryptographic guarantees. Loans of crypto-currencies must use some form of multi-signature with an escrow agent, so that when loans need to be recalled, they can be done so fully and systematically without creating risk. Similarly the practice of commingling assets must be banned if we want to achieve stability in the face of market movements.

In conclusion, the only form of sensible stablecoin or token representing another asset without systemic risk is the fully collateralized kind. The reason to create such a stablecoin is to enhance fiat currencies and imbue them with the cryptographic certainty, fast settlement time, and international transportability of crypto-currencies. It would make a lot of sense for the Federal Reserve or European Central Bank to be the one to issue such a bond. It is extremely important that the collateral behind the stablecoin cannot be rehypothecated, and that the collateral is actually in the underlying, and not commingled with other assets having their own market movements. We encourage regulators to focus on the topics of rehypothecation and commingling so as to not bring the failure modes of traditional finance into our crypto-financial future.

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